The Generalized Inverse Sum Indeg Index of Some Graph Operations

Abstract

The study of networks and graphs carried out by topological measures performs a vital role in securing their hidden topologies. This strategy has been extremely used in biomedicine, cheminformatics and bioinformatics, where computations dependent on graph invariants have been made available to communicate the various challenging tasks. In quantitative structure–activity (QSAR) and quantitative structure–property (QSPR) relationship studies, topological invariants are brought into practical action to associate the biological and physicochemical properties and pharmacological activities of materials and chemical compounds. In these studies, the degree-based topological invariants have found a significant position among the other descriptors due to the ease of their computing process and the speed with which these computations can be performed. Thereby, assessing these invariants is one of the flourishing lines of research. The generalized form of the degree-based inverse sum indeg index has recently been introduced. Many degree-based topological invariants can be derived from the generalized form of this index. In this paper, we provided the bounds related to this index for some graph operations, including the Kronecker product, join, corona product, Cartesian product, disjunction, and symmetric difference. We also presented the exact formula of this index for the disjoint union, linking, and splicing of graphs.

1. Introduction

Cheminformatics is a novel information technology field involving Biology, Chemistry, Biochemistry, Physics, Statistics, Mathematics, and other informational sciences that focus on storing, gathering, examining, and treating chemical data. Cheminformatics is a significant application of information technology to help chemists investigate new problems and understand, analyse, and organise scientific data in developing novel materials, compounds, and processes [1,2]. Chemical and molecular graphs have fundamental applications in chemoinformatics, quantitative structure–activity (QSAR) and quantitative structure–property (QSPR) relationship studies, computational drug design, and virtual screening of chemical libraries. Chemoinformatics applications of graphs includes coding and chemical structure illustration, physicochemical property prediction, and database searching and retrieval. Virtual screening, QSPRs, and QSARs are established on the structure–property principle, which says that chemical compounds’ biological and physicochemical properties can be forecasted from their chemical structure. Such structure–property linkages are usually developed from topological invariants, fingerprints calculated from the molecular graph, and molecular descriptors calculated from the three-dimensional chemical structure [3,4,5,6]. Chemical graph theory is a branch of mathematical chemistry that analyses the physical and chemical properties (heat of formation and solubility, surface tension and density, acentric factor, motor octane number, octanol–water partition coefficient, boiling points, freezing points, melting points, molar refraction) of the chemical graphs with the assistance of mathematical models which are designed by the graph-theoretic approaches of the subject of graph theory. Cheminformatics deals with a chemical phenomenon known as the QSPRs and QSARs of chemical compounds. Cheminformatics techniques are also utilized for forecasting properties applicable to optimization and drug discovery. For instance, knowledge discovery can be used to identify lead compounds in pharmaceutical data matching. This era is devoted to improving chemical sciences, such as bond formation in chemical compounds, drug discovery, and developing diagnostic kits for various biological procedures and diseases. All these developments need tools to make these improvements achievable to move the way they are thrusting. An emerging tool in investigating these phenomena is a topological index that remains invariant for all chemical networks up to their isomorphisms. A topological index of a chemical graph is a change in its molecular structure into some real number and describes the topology of the underlying graph. Many topological indices can be used in QSAR/QSPR analysis [3,5], which makes conclusions about the physicochemical and bioactive properties of underlying structures. Different types of spectral, degree-based, distance-based, and polynomial-related indices of graphs are successfully used and widely computed [7,8,9,10,11,12,13,14,15,16,17,18]. Out of these classes, degree-based indices become the most valuable and play a remarkable role in chemical graph theory. These invariants are extensively used in combination to infer pharmacological, biological, and physicochemical properties such as the critical temperature, density, stability, surface tension, chirality, enthalpy of formation, entropy, boiling point, melting point, connectivity, similarity, toxicity, etc., of chemical compounds in chemical graph theory [19,20,21].

Let $\mathcal{T}$ be a graph with an edge set $E\left(\mathcal{T}\right)$ and a vertex set $V\left(\mathcal{T}\right)$. For any $a\in V\left(\mathcal{T}\right)$, $N_{\mathcal{T}}\left(a\right)$ is the set of all vertices adjacent to a in graph $\mathcal{T}$. The elements of $N_{\mathcal{T}}\left(a\right)$ are called neighbors of a. The number of neighbors of $a\in \mathcal{T}$ are said to be the degree of a, represented by $d_{\mathcal{T}}\left(a\right)$. For the graph $\mathcal{T},$ its order $n_{\mathcal{T}}$ is $n_{\mathcal{T}}=\left|V\left(\mathcal{T}\right)\right|$, and its size $m_{\mathcal{T}}$ is $m_{\mathcal{T}}=\left|E\left(\mathcal{T}\right)\right|$. The smallest (respectively, largest) degree of $\mathcal{T}$ is the smallest (respectively, largest) vertex degree in $\mathcal{T}$, denoted as ${\delta }_{\mathcal{T}}$ (respectively, ${\Delta }_{\mathcal{T}}$). Throughout this paper, all the graphs will be simple and connected.

Wiener introduced the first topological invariant (1947) [22], which is related to the density of paraffin, critical points, and boiling points. Randic presented the Randic index in 1975 [15] and generalized it by Bollob$\acute{\mathrm{a}}$s and Erd$\ddot{\mathrm{o}}$s (1998) [23]. Gutman and Trinajstic (1972) introduced the first and second Zagreb indices [12]. The atom-bond connectivity (ABC) index was presented by Estrada et al. (1998) [24], and the geometric-arithmetic index was proposed by Vukičević and Furtula (2009) [17]. In 2010, Vukičević and Gašperov [25] defined the set of 148 discrete adriatic indices. Against the International Academy of Mathematical Chemistry testing sets, these indices showed good predictive properties. Twenty of these descriptors were taken as noteworthy predictors of physicochemical properties. Among these indices, the inverse sum indeg (ISI) index is proposed as an appreciable forecaster of total surface area for octane isomers [25] and is expressed as:

$$ISI\left(\mathcal{T}\right)=\sum _{wz\in E\left(\mathcal{T}\right)}\frac{d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)}{d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)}.$$

In the literature, numerous papers find a closed formula for a degree-based topological invariant for a family of graphs. The generalizations and modification of the standard degree-based topological invariants are introduced to overcome this specific approach. This approach may improve the existing outcomes and correlate best to several physicochemical properties of underlying structures. Due to these motivating reasons, Hafeez and Farooq [26] proposed the generalized form of the ISI index as follows:

$$S_{\alpha ,\beta }\left(\mathcal{T}\right)=\sum _{wz\in E\left(\mathcal{T}\right)}\frac{{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }},$$

(1)

where $\alpha$ and $\beta$ are real numbers. They demonstrated that many degree-based topological indices could be derived from the generalized ISI index by giving the specific values to the parameters $\alpha$ and $\beta$. By knowing the generalized ISI index of a given family of graphs, the precise formula for any such index can be acquired routinely. Knowing the generalized ISI index is sufficient, future research should focus on finding the generalized ISI index of a family of graphs rather than finding the related indices one by one. Moreover, we hope that a closer look at the properties and results of the generalized ISI index of graph operations will bring new general insights. Buragohain et al. [8] computed the generalized ISI index of some widely used chemical structures which often appear in chemical graph theory. The first general Zagreb index of a graph $\mathcal{T}$ is introduced by Li and Zhao [27] and is defined as follows:

$$M_1^{\alpha }\left(\mathcal{T}\right)=\sum _{w\in V\left(\mathcal{T}\right)}{\left(d_{\mathcal{T}}\left(w\right)\right)}^{\alpha }.$$

The general Randić index was proposed by Li and Gutman [28] and is defined as follows:

$$R_{\alpha }\left(\mathcal{T}\right)=\sum _{wz\in E\left(\mathcal{T}\right)}{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }.$$

Zhou and Trinajstić [29] gave the expression of the general sum connectivity index as:

$${\chi }_{\alpha }\left(\mathcal{T}\right)=\sum _{wz\in E\left(\mathcal{T}\right)}{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right))}^{\alpha }.$$

Graph operations are operations which construct new graphs from initial ones. Complex networks or large molecular structures can be formed by applying some graph operations on simple graphs. In addition, these simple graphs can assist in explaining some of the properties of these structures. The studies of graph products have been intensively carried out in the last few decades, and by now, a rich theory for the recognition and structure of classes of these graphs has become visible [30]. In chemical graph theory, graph operations performed a remarkable role, as some chemically interesting graphs can be constructed by different graph operations on some particular or general graphs. Graph operations also carried out a vital task in studying the decomposition of graphs into isomorphic subgraphs. For the characteristics of some nanostructures and molecular graphs, a powerful tool is the correlations attained for various attributes of graph operations in the form of characteristics of their respective components. For the results related to different topological descriptors under graph operations, we refer the interested reader to [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].

In the next section, we find bounds for some graph operations, including the Kronecker product, join, the corona product, the Cartesian product, disjunction, and symmetric difference. Section 3 contains the result, in which we find the exact formula of the generalized ISI index for disjoint union, link, and splicing of graphs.

2. Bounds on ${\mathbf{S}}_{\mathbf{\alpha },\mathbf{\beta }}\left(\mathcal{T}\right)$ for Some Graph Operations

In this section, we give bounds on the generalized ISI index for the Kronecker product, join, the corona product, the Cartesian product, disjunction, and the symmetric difference of graphs.

2.1. Kronecker Product of Graphs

The Kronecker product $\mathcal{T}\oslash \mathcal{H}$ of graphs $\mathcal{T}$ and $\mathcal{H}$ is the graph with vertex set $V\left(\mathcal{T}\right)\times V\left(\mathcal{H}\right)$ and $(w_1,z_1)(w_2,z_2)\in E(\mathcal{T}\oslash \mathcal{H})$ if and only if $w_1{w}_2\in E\left(\mathcal{T}\right)$ and $z_1{z}_2\in E\left(\mathcal{H}\right)$. For any $(w,z)\in V(\mathcal{T}\oslash \mathcal{H})$, we have:

$$d_{\mathcal{T}\oslash \mathcal{H}}\left((w,z)\right)=d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right).$$

(2)

In the next theorem, we compute the bounds for the Kronecker product of two graphs for the generalized ISI index.

Theorem 1.

For the graphs$\mathcal{T}$and$\mathcal{H},$we have the following:

(1). 

If$\alpha \in \mathbb{R}$and$\beta \ge 0$, then$\frac{R_{\alpha }\left(\mathcal{T}\right)R_{\alpha }\left(\mathcal{H}\right)}{2^{\beta -1}{\left({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}\right)}^{\beta }}\le S_{\alpha ,\beta }(\mathcal{T}\oslash \mathcal{H})\le \frac{R_{\alpha }\left(\mathcal{T}\right)R_{\alpha }\left(\mathcal{H}\right)}{2^{\beta -1}{\left({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}\right)}^{\beta }}$.

(2). 

If$\alpha \in \mathbb{R}$and$\beta \le 0$, then$\frac{R_{\alpha }\left(\mathcal{T}\right)R_{\alpha }\left(\mathcal{H}\right)}{2^{\beta -1}{\left({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}\right)}^{\beta }}\le S_{\alpha ,\beta }(\mathcal{T}\oslash \mathcal{H})\le \frac{R_{\alpha }\left(\mathcal{T}\right)R_{\alpha }\left(\mathcal{H}\right)}{2^{\beta -1}{\left({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}\right)}^{\beta }}$.

In both cases, the inequality becomes an equality if$\mathcal{T}$and$\mathcal{H}$are regular graphs.

Proof. 

$\left(1\right).$ Let $\alpha \in \mathbb{R}$ and $\beta \ge 0$. Then, for any $w\in V\left(\mathcal{T}\right)$, $d_{\mathcal{T}}\left(w\right)\le {\Delta }_{\mathcal{T}}$ and $d_{\mathcal{T}}\left(w\right)\ge {\delta }_{\mathcal{T}}$. Hence, from Equation (2) and definition of the generalized ISI index, we obtain:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\oslash \mathcal{H})& =& 2\sum _{uw\in E\left(\mathcal{T}\right)}\sum _{vz\in E\left(\mathcal{H}\right)}\left(\frac{{\left(d_{\mathcal{T}}\left(u\right)d_{\mathcal{H}}\left(v\right)d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(u\right)d_{\mathcal{H}}\left(v\right)+d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ & \ge & 2\sum _{uw\in E\left(\mathcal{T}\right)}\sum _{vz\in E\left(\mathcal{H}\right)}\left(\frac{{\left(d_{\mathcal{T}}\left(u\right)d_{\mathcal{H}}\left(v\right)d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{2^{\beta }{\left({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}\right)}^{\beta }}\right)\hfill \\ & =& \frac{1}{2^{\beta -1}{\left({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}\right)}^{\beta }}\sum _{uw\in E\left(\mathcal{T}\right)}{\left(d_{\mathcal{T}}\left(u\right)d_{\mathcal{T}}\left(w\right)\right)}^{\alpha }\sum _{vz\in E\left(\mathcal{H}\right)}{\left(d_{\mathcal{H}}\left(v\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }\hfill \\ & =& \frac{R_{\alpha }\left(\mathcal{T}\right)R_{\alpha }\left(\mathcal{H}\right)}{2^{\beta -1}{\left({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}\right)}^{\beta }}.\hfill \end{array}$$

The equality holds if $\mathcal{T}$ and $\mathcal{H}$ are regular graphs. Similarly, we can obtain $S_{\alpha ,\beta }(\mathcal{T}\oslash \mathcal{H})\le \frac{R_{\alpha }\left(\mathcal{T}\right)R_{\alpha }\left(\mathcal{H}\right)}{2^{\beta -1}{\left({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}\right)}^{\beta }}$. Part $\left(2\right)$ can be proven in a similar manner. □

Corollary 1.

Let$\mathcal{T}$be a k-regular graph having size$m_{\mathcal{T}}.$Then$S_{\alpha ,\beta }(\mathcal{T}\oslash \mathcal{H})=2m_{\mathcal{T}}{k}^{2\alpha -\beta }{S}_{\alpha ,\beta }\left(\mathcal{H}\right)$, where α and β are real numbers.

2.2. Join of Graphs

For the graphs $\mathcal{T}$ and $\mathcal{H}$, the join $\mathcal{T}\vee \mathcal{H}$ is formed from $\mathcal{T}\cup \mathcal{H}$ (the disjoint union $\mathcal{T}\cup \mathcal{H}$ is the graph with $V(\mathcal{T}\cup \mathcal{H})=V\left(\mathcal{T}\right)\cup V\left(\mathcal{H}\right)$ and $E(\mathcal{T}\cup \mathcal{H}=E(\mathcal{T})\cup E(\mathcal{H})$) and by making every vertex of $\mathcal{T}$ adjacent to every vertex of $\mathcal{H}$. For any $z\in V(\mathcal{T}\vee \mathcal{H})$, we have:

$$d_{(\mathcal{T}\vee \mathcal{H})}\left(z\right)=\left\{\begin{array}{cc}{d}_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}}\hfill & \mathrm{if} z\in V\left(\mathcal{T}\right),\hfill \\ d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}\hfill & \mathrm{if} z\in V\left(\mathcal{H}\right).\hfill \end{array}\right.$$

(3)

Theorem 2.

Suppose$\mathcal{T}$and$\mathcal{H}$are two graphs. Then,

(1). 

If$\alpha ,\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \ge & \frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{{(m_{\mathcal{T}}+1+2n_{\mathcal{H}})}^{\beta }}+\frac{R_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{H}}{n}_{\mathcal{T}}^{2\alpha }}{{(m_{\mathcal{H}}+1+2n_{\mathcal{T}})}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{M}_1^{\alpha }\left(\mathcal{T}\right)M_1^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{H}}{n}_{\mathcal{T}}^{\alpha }{M}_1^{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{T}}{n}_{\mathcal{H}}^{\alpha }{M}_1^{\alpha }\left(\mathcal{H}\right)+{\left(n_{\mathcal{T}}{n}_{\mathcal{H}}\right)}^{\alpha +1}}{{(2n_{\mathcal{T}}+2n_{\mathcal{H}}-2)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+n_{\mathcal{T}})}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }},\hfill \end{array}$$

(2). 

If$\alpha \ge 0$and$\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{{(m_{\mathcal{T}}+2n_{\mathcal{H}}+1)}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{{(m_{\mathcal{H}}+2n_{\mathcal{T}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{T}}+2n_{\mathcal{H}}-2)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \ge & \frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{R_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{H}}{n}_{\mathcal{T}}^{2\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+n_{\mathcal{T}})}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{M}_1^{\alpha }\left(\mathcal{T}\right)M_1^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{n}_{\mathcal{H}}{M}_1^{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{n}_{\mathcal{T}}{M}_1^{\alpha }\left(\mathcal{H}\right)+{\left(n_{\mathcal{T}}{n}_{\mathcal{H}}\right)}^{\alpha +1}}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(3). 

If$\alpha \le 0$and$\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \ge & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{{(m_{\mathcal{T}}+2n_{\mathcal{H}}+1)}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{{(m_{\mathcal{H}}+2n_{\mathcal{T}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{T}}+2n_{\mathcal{H}}-2)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\delta }_{\mathcal{H}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+n_{\mathcal{H}})}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}+n_{\mathcal{T}}{\delta }_{\mathcal{T}}+n_{\mathcal{H}}{\delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\delta }_{\mathcal{H}}+n_{\mathcal{H}}^2)}^{\alpha }}{{(m_{\mathcal{T}}+2n_{\mathcal{H}}+1)}^{\beta }}+\frac{m_{\mathcal{H}}{({\delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{{(m_{\mathcal{H}}+2n_{\mathcal{T}}+1)}^{\beta }}\hfill \end{array}$$

$$\begin{array}{ccc}& +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}+n_{\mathcal{T}}{\delta }_{\mathcal{T}}+n_{\mathcal{H}}{\delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{T}}+2n_{\mathcal{H}}-2)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& \ge & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+n_{\mathcal{T}})}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

In every case, the inequality becomes equality if$\mathcal{T}$and$\mathcal{H}$are regular graphs.

Proof. 

$\left(1\right).$ Let $\alpha ,\beta \ge 0$. From the definition of the generalized ISI index and Equation (3), we obtain:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\vee \mathcal{H})& =& \sum _{wz\in E(\mathcal{T}\vee \mathcal{H})}\left(\frac{{\left(d_{\mathcal{T}\vee \mathcal{H}}\left(w\right)d_{\mathcal{T}\vee \mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}\vee \mathcal{H}}\left(w\right)+d_{\mathcal{T}\vee \mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ & =& \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & +& \sum _{wz\in E\left(\mathcal{H}\right)}\left(\frac{{(d_{\mathcal{H}}\left(w\right)+n_{\mathcal{T}})}^{\alpha }{(d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}})}^{\alpha }}{{(d_{\mathcal{H}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+2n_{\mathcal{T}})}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}w\in V\left(\mathcal{T}\right)\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& Q_1+Q_2+Q_3,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill Q_1& =& \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+2n_{\mathcal{H}})}^{\beta }}\right),\hfill \\ \hfill Q_2& =& \sum _{wz\in E\left(\mathcal{H}\right)}\left(\frac{{(d_{\mathcal{H}}\left(w\right)+n_{\mathcal{T}})}^{\alpha }{(d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}})}^{\alpha }}{{(d_{\mathcal{H}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+2n_{\mathcal{T}})}^{\beta }}\right),\hfill \\ \hfill Q_3& =& \sum _{\begin{array}{c}w\in V\left(\mathcal{T}\right)\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}\right).\hfill \end{array}$$

For any $wz\in E\left(\mathcal{T}\right)$, we have $d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)\le m_{\mathcal{T}}+1$. Also for $x,y,t>0$, ${(x+y)}^t\ge x^t+y^t$. Therefore,

$$\begin{array}{ccc}\hfill Q_1& =& \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & \ge & \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }+n_{\mathcal{H}}^{\alpha }{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right))}^{\alpha }+n_{\mathcal{H}}^{2\alpha }}{{(m_{\mathcal{T}}+1+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{{(m_{\mathcal{T}}+1+2n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

In addition,

$$\begin{array}{ccc}\hfill Q_1& =& \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{({\Delta }_{\mathcal{T}}+n_2)}^{\alpha }{({\Delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\alpha }}{{(2{\delta }_{\mathcal{T}}+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

Therefore,

$$\frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{{(m_{\mathcal{T}}+1+2n_{\mathcal{H}})}^{\beta }}\le Q_1\le \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.$$

(4)

Similarly,

$$\frac{R_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{H}}{n}_{\mathcal{T}}^{2\alpha }}{{(m_{\mathcal{H}}+1+2n_{\mathcal{T}})}^{\beta }}\le Q_2\le \frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+n_{\mathcal{T}})}^{\beta }}.$$

(5)

Since $\mathcal{T}$ is connected, the degree of any vertex of $\mathcal{T}$ is at most $n_{\mathcal{T}}-1$. Now, for $Q_3$, we have:

$$\begin{array}{cc}\hfill Q_3& =\sum _{\begin{array}{c}w\in V\left(\mathcal{T}\right)\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{(d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}{d}_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}}{d}_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ \hfill & \ge \sum _{\begin{array}{c}w\in V\left(\mathcal{T}\right)\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }+{\left(n_{\mathcal{T}}{d}_{\mathcal{T}}\left(w\right)\right)}^{\alpha }+{\left(n_{\mathcal{H}}{d}_{\mathcal{H}}\left(z\right)\right)}^{\alpha }+{\left(n_{\mathcal{T}}{n}_{\mathcal{H}}\right)}^{\alpha }}{{((n_{\mathcal{T}}-1)+(n_{\mathcal{H}}-1)+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ \hfill & =\frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{M}_1^{\alpha }\left(\mathcal{T}\right)M_1^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{H}}{n}_{\mathcal{T}}^{\alpha }{M}_1^{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{T}}{n}_{\mathcal{H}}^{\alpha }{M}_1^{\alpha }\left(\mathcal{H}\right)+{\left(n_{\mathcal{T}}{n}_{\mathcal{H}}\right)}^{\alpha +1}}{{(2n_{\mathcal{T}}+2n_{\mathcal{H}}-2)}^{\beta }}.\hfill \end{array}$$

(6)

and

$$\begin{array}{ccc}\hfill Q_3& =& \sum _{\begin{array}{c}w\in V\left(\mathcal{T}\right)\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{(d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}{d}_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}}{d}_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & \le & \sum _{\begin{array}{c}w\in V\left(\mathcal{T}\right)\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(7)

Now adding the Equations (4)–(7), we obtain the required result. The equality holds if $\mathcal{T}$ and $\mathcal{H}$ are regular graphs.

$\left(2\right).$ Let $\alpha \ge 0$ and $\beta \le 0$. As $\beta \le 0$, therefore, for any $wz\in E\left(\mathcal{T}\right)$, we have ${(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }\ge {(m_{\mathcal{T}}+1)}^{\beta }$. Then,

$$\begin{array}{ccc}\hfill Q_1& =& \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & \le & \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{{(m_{\mathcal{T}}+1+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{{(m_{\mathcal{T}}+1+2n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(8)

Similarly, one can show that

$$\begin{array}{ccc}\hfill Q_2& \le & \frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}^2)}^{\alpha }}{{(m_{\mathcal{H}}+1+2n_{\mathcal{T}})}^{\beta }},\hfill \\ \hfill Q_3& \le & \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{T}}{n}_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{T}}+2n_{\mathcal{H}}-2)}^{\beta }}.\hfill \end{array}$$

(9)

Now,

$$\begin{array}{ccc}\hfill Q_1& =& \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{(d_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}})}^{\alpha }{(d_{\mathcal{T}}\left(z\right)+n_{\mathcal{H}})}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & \ge & \sum _{wz\in E\left(\mathcal{T}\right)}\left(\frac{{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }+n_{\mathcal{H}}^{\alpha }{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right))}^{\alpha }+n_{\mathcal{H}}^{2\alpha }}{{(2{\delta }_{\mathcal{T}}+2n_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(10)

Similarly, one can prove that

$$\begin{array}{ccc}\hfill Q_2& \ge & \frac{R_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{H}}{n}_{\mathcal{T}}^{2\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+n_{\mathcal{T}})}^{\beta }}\hfill \\ \hfill Q_3& \ge & \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}{M}_1^{\alpha }\left(\mathcal{T}\right)M_1^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{n}_{\mathcal{H}}{M}_1^{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{n}_{\mathcal{T}}{M}_1^{\alpha }\left(\mathcal{H}\right)+{\left(n_{\mathcal{T}}{n}_{\mathcal{H}}\right)}^{\alpha +1}}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(11)

By adding Equation (8) to (9) and Equation (10) to (11), the desired result is obtained. The equality holds if $\mathcal{T}$ and $\mathcal{H}$ are regular graphs. In a similar manner, one can prove Parts $\left(3\right)$ and $\left(4\right)$. □

Example 1.

The fan graph$F_{n+1}$is the join of$K_1$with$P_n$, and the wheel graph$W_{n+1}$is the join of$K_1$with$C_n$(seeFigure 1). Using the Theorem 2, the following bounds are obtained for$S_{\alpha ,\beta }\left(F_{n+1}\right)$:

(1). 

If$\alpha ,\beta \ge 0$,m then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }\left(F_{n+1}\right)& \ge & \frac{4^{\alpha }(2n-6)+2(2^{\alpha }+3^{\alpha })+n-1}{{(2+n)}^{\beta }}+\frac{n^{\alpha -\beta }(n+2+2^{\alpha }(n-2))}{2^{\beta }}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }\left(F_{n+1}\right)& \le & \frac{9^{\alpha }(n-1)}{4^{\beta }}+\frac{3^{\alpha }{n}^{\alpha +1}}{{(2+n)}^{\beta }},\hfill \end{array}$$

(2). 

If$\alpha \ge 0$and$\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }\left(F_{n+1}\right)& \ge & \frac{4^{\alpha }(2n-6)+2(2^{\alpha }+3^{\alpha })+n-1}{{(2+n)}^{\beta }}+\frac{n^{\alpha -\beta }(n+2+2^{\alpha }(n-2))}{2^{\beta }}\hfill \\ \hfill S_{\alpha ,\beta }\left(F_{n+1}\right)& \le & \frac{9^{\alpha }(n-1)}{{(2+n)}^{\beta }}+\frac{3^{\alpha }{n}^{\alpha +1}}{{\left(2n\right)}^{\beta }},\hfill \end{array}$$

(3). 

If$\alpha \le 0$and$\beta \ge 0$, then

$$\frac{9^{\alpha }(n-1)}{{(2+n)}^{\beta }}+\frac{3^{\alpha }{n}^{\alpha +1}}{{\left(2n\right)}^{\beta }}\le S_{\alpha ,\beta }\left(F_{n+1}\right)\le \frac{4^{\alpha }(n-1)}{2^{\beta }{(1+n)}^{\beta }}+\frac{2^{\alpha }{n}^{\alpha +1}}{{(2+n)}^{\beta }},$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\frac{9^{\alpha }(n-1)}{{\left(4\right)}^{\beta }}+\frac{2^{\alpha }{n}^{\alpha +1}}{{(2+n)}^{\beta }}\le S_{\alpha ,\beta }\left(F_{n+1}\right)\le \frac{4^{\alpha }(n-1)}{{(2+n)}^{\beta }}+\frac{2^{\alpha }{n}^{\alpha +1}}{{\left(2n\right)}^{\beta }}.$$

We have to compare these above results with actual values: for example, when$n=4,$$\alpha =2$and$\beta =3$, we obtain$S_{2,3}\left(F_{4+1}\right)=2.3832$, that is,$0.72917<2.3832<6.4635$. Similarly, one can obtain the bounds on$S_{\alpha ,\beta }\left(W_{n+1}\right)$.

2.3. Corona Product of Graphs

For the graphs $\mathcal{T}$ and $\mathcal{H},$ the corona product $\mathcal{T}\circ \mathcal{H}$ is formed by taking $n_{\mathcal{T}}$ copies of $\mathcal{H}$ and one copy of $\mathcal{T}$ and by making the j-th vertex of $\mathcal{T}$ adjacent to every vertex of the j-th copy of $\mathcal{H}$, where $1\le j\le n_{\mathcal{T}}$. For any $w\in V(\mathcal{T}\circ \mathcal{H})$, we have:

$$d_{(\mathcal{T}\circ \mathcal{H})}\left(w\right)=\left\{\begin{array}{cc}{d}_{\mathcal{T}}\left(w\right)+n_{\mathcal{H}}\hfill & \mathrm{if} w\in V\left(\mathcal{T}\right),\hfill \\ d_{\mathcal{H}}\left(w\right)+1\hfill & \mathrm{if} w\in V\left({\mathcal{H}}_j\right),1\le j\le n_{\mathcal{T}},\hfill \end{array}\right.$$

(12)

where ${\mathcal{H}}_j$ is the j-th copy of $\mathcal{H}$.

By using the Equations (1) and (12) and all the facts used in Theorem 2, one can obtain the following theorem.

Theorem 3.

Suppose$\mathcal{T}$and$\mathcal{H}$are two graphs. Then:

(1). 

If$\alpha ,\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \ge & \frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{{(2{\Delta }_{\mathcal{T}}+2n_{\mathcal{H}})}^{\beta }}+\frac{n_{\mathcal{T}}(R_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{H}}{n}_{\mathcal{T}}^{2\alpha })}{{(2{\Delta }_{\mathcal{H}}+2)}^{\beta }}\hfill \\ & +& \frac{{M_1}^{\alpha }\left(\mathcal{T}\right){M_1}^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{H}}{M_1}^{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{T}}{n}_{\mathcal{H}}^{\alpha }{M_1}^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}{n}_{\mathcal{H}}^{\alpha +1}}{{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{H}}+1)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{H}})}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }},\hfill \end{array}$$

(2). 

If$\alpha \ge 0$and$\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \ge & \frac{R_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{T}\right)+m_{\mathcal{T}}{n}_{\mathcal{H}}^{2\alpha }}{{(2{\delta }_{\mathcal{T}}+2n_{\mathcal{H}})}^{\beta }}+\frac{n_{\mathcal{T}}(R_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}^{\alpha }{\chi }_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{H}}{n}_{\mathcal{T}}^{2\alpha })}{{(2{\delta }_{\mathcal{H}}+2)}^{\beta }}\hfill \\ & +& \frac{{M_1}^{\alpha }\left(\mathcal{T}\right){M_1}^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{H}}{M_1}^{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{T}}{n}_{\mathcal{H}}^{\alpha }{M_1}^{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}{n}_{\mathcal{H}}^{\alpha +1}}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{H}}+1)}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{H}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{H}})}{{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }},\hfill \end{array}$$

(3). 

If$\alpha \le 0$ and $\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \ge & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{H}}+1)}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{H}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{H}})}{{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{H}}+1)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}+{\delta }_{\mathcal{T}}+n_{\mathcal{H}}{\delta }_{\mathcal{H}}+n_{\mathcal{H}})}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }}.\hfill \end{array}$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \ge & \frac{m_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{H}}+1)}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{H}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}({\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}}+{\Delta }_{\mathcal{T}}+n_{\mathcal{H}}{\Delta }_{\mathcal{H}}+n_{\mathcal{H}})}{{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\circ \mathcal{H})& \le & \frac{m_{\mathcal{T}}{({\delta }_{\mathcal{T}}^2+2n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{H}}^2)}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+n_{\mathcal{H}})}^{\beta }}+\frac{m_{\mathcal{H}}{({\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{H}}+1)}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{H}}+1)}^{\beta }}\hfill \\ & +& \frac{n_{\mathcal{T}}{n}_{\mathcal{H}}({\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}}+{\delta }_{\mathcal{T}}+n_{\mathcal{H}}{\delta }_{\mathcal{H}}+n_{\mathcal{H}})}{{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}}+n_{\mathcal{H}}+1)}^{\beta }}.\hfill \end{array}$$

Example 2.

The bottleneck graph B of a graph$\mathcal{T}$, where$B=K_2\circ \mathcal{T}$. One can obtain the bounds for this graph by using the Theorem 3.

2.4. Cartesian Product of Graphs

For the graphs $\mathcal{T}$ and $\mathcal{H}$, the Cartesian product $\mathcal{T}\square \mathcal{H}$ is the graph with vertex set $V\left(\mathcal{T}\right)\times V\left(\mathcal{H}\right)$ and $(w_1,z_1)(w_2,z_2)\in E(\mathcal{T}\square \mathcal{H})$ if and only if $w_1=w_2$ and $z_1{z}_2\in E\left(\mathcal{H}\right)$ or $z_1=z_2$ and $w_1{w}_2\in E\left(\mathcal{T}\right)$. For any $(w,z)\in V(\mathcal{T}\square \mathcal{H})$,

$$d_{\mathcal{T}\square \mathcal{H}}\left((w,z)\right)=d_{\mathcal{T}}\left(w\right)+d_{\mathcal{H}}\left(z\right).$$

(13)

Theorem 4.

For the graphs$\mathcal{T}$and$\mathcal{H},$we have:

(1). 

If$\alpha ,\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})& \ge & \frac{1}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}(m_{\mathcal{H}}{M}_1^{2\alpha }\left(\mathcal{T}\right)+M_1^{\alpha }\left(\mathcal{T}\right){\chi }_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}{R}_{\alpha }\left(\mathcal{H}\right)\hfill \\ & +& m_{\mathcal{T}}{M}_1^{2\alpha }\left(\mathcal{H}\right)+M_1^{\alpha }\left(\mathcal{H}\right){\chi }_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}{R}_{\alpha }\left(\mathcal{T}\right))\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})& \le & \frac{(n_{\mathcal{T}}{m}_{\mathcal{H}}+m_{\mathcal{T}}{n}_{\mathcal{H}}){({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(2). 

If$\alpha \ge 0$and$\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})& \ge & \frac{1}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}(m_{\mathcal{H}}{M}_1^{2\alpha }\left(\mathcal{T}\right)+M_1^{\alpha }\left(\mathcal{T}\right){\chi }_{\alpha }\left(\mathcal{H}\right)\hfill \\ & +& n_{\mathcal{T}}{R}_{\alpha }\left(\mathcal{H}\right)+m_{\mathcal{T}}{M}_1^{2\alpha }\left(\mathcal{H}\right)+M_1^{\alpha }\left(\mathcal{H}\right){\chi }_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}{R}_{\alpha }\left(\mathcal{T}\right)),\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})& \le & \frac{(n_{\mathcal{T}}{m}_{\mathcal{H}}+m_{\mathcal{T}}{n}_{\mathcal{H}}){({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(3). 

If$\alpha \le 0$and$\beta \ge 0$, then

$$\begin{array}{ccc}\hfill \frac{(n_{\mathcal{T}}{m}_{\mathcal{H}}+m_{\mathcal{T}}{n}_{\mathcal{H}}){({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}& \le & S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})\hfill \\ & \le & \frac{(n_{\mathcal{T}}{m}_{\mathcal{H}}+m_{\mathcal{T}}{n}_{\mathcal{H}}){({\delta }_{\mathcal{T}}^2+{\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\begin{array}{ccc}\hfill \frac{(n_{\mathcal{T}}{m}_{\mathcal{H}}+m_{\mathcal{T}}{n}_{\mathcal{H}}){({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}& \le & S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})\hfill \\ & \le & \frac{(n_{\mathcal{T}}{m}_{\mathcal{H}}+m_{\mathcal{T}}{n}_{\mathcal{H}}){({\delta }_{\mathcal{T}}^2+{\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

Proof. 

$\left(1\right).$ Let $\alpha ,\beta \ge 0$. From the definition of the generalized ISI index and Equation (13), we obtain:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\square \mathcal{H})& =& \sum _{(w_1,z_1)(w_2,z_2)\in E(\mathcal{T}\square \mathcal{H})}\left(\frac{{\left(d_{\mathcal{T}\square \mathcal{H}}\left((w_1,z_1)\right)d_{\mathcal{T}\square \mathcal{H}}\left((w_2,z_2)\right)\right)}^{\alpha }}{{(d_{\mathcal{T}\square \mathcal{H}}\left((w_1,z_1)\right)+d_{\mathcal{T}\square \mathcal{H}}\left((w_2,z_2)\right))}^{\beta }}\right)\hfill \\ & =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right))(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}{z}_1\in V\left(\mathcal{H}\right)\\ w_1{w}_2\in E\left(\mathcal{T}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{T}}\left(w_1\right))(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{T}}\left(w_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))}^{\beta }}\right)\hfill \\ & =& Y_1+Y_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill Y_1& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right))(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right),\hfill \\ \hfill Y_2& =& \sum _{\begin{array}{c}{z}_1\in V\left(\mathcal{H}\right)\\ w_1{w}_2\in E\left(\mathcal{T}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{T}}\left(w_1\right))(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{T}}\left(w_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))}^{\beta }}\right).\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill Y_1& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right))(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \ge & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left(d_{\mathcal{T}}\left(w_1\right)\right)}^{2\alpha }+{\left(d_{\mathcal{T}}\left(w_1\right)(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }+{\left(d_{\mathcal{H}}\left(z_1\right)d_{\mathcal{H}}\left(z_2\right)\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \ge & \frac{m_{\mathcal{H}}{M}_1^{2\alpha }\left(\mathcal{T}\right)+M_1^{\alpha }\left(\mathcal{T}\right){\chi }_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}{R}_{\alpha }\left(\mathcal{H}\right)}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Y_1& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right))(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \le & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}{m}_{\mathcal{H}}\left({({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }\right)}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

Therefore,

$$\frac{m_{\mathcal{H}}{M}_1^{2\alpha }\left(\mathcal{T}\right)+M_1^{\alpha }\left(\mathcal{T}\right){\chi }_{\alpha }\left(\mathcal{H}\right)+n_{\mathcal{T}}{R}_{\alpha }\left(\mathcal{H}\right)}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}\le Y_1\le \frac{n_{\mathcal{T}}{m}_{\mathcal{H}}\left({({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }\right)}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}.$$

(14)

Similarly, one can prove that

$$\begin{array}{ccc}\hfill \frac{m_{\mathcal{T}}{M}_1^{2\alpha }\left(\mathcal{H}\right)+M_1^{\alpha }\left(\mathcal{H}\right){\chi }_{\alpha }\left(\mathcal{T}\right)+n_{\mathcal{H}}{R}_{\alpha }\left(\mathcal{T}\right)}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }},& \le & Y_2\le \frac{n_{\mathcal{H}}{m}_{\mathcal{T}}{({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(15)

By adding the Equations (14) and (15), the desired result is obtained. Analogously, one can prove Parts $\left(2\right)$ and $\left(3\right)$.

$\left(4\right).$ Let $\alpha ,\beta \le 0$. Then, for any vertex $w\in V\left(\mathcal{T}\right)$, ${\Delta }_{\mathcal{T}}^{\alpha }\le {\left(d_{\mathcal{T}}\left(w\right)\right)}^{\alpha }\le {\delta }_{\mathcal{T}}^{\alpha }$. Therefore,

$$\begin{array}{ccc}\hfill Y_1& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right))(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& \ge & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}{m}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Y_1& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right))(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(2d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \le & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{({\delta }_{\mathcal{T}}^2+{\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}{m}_{\mathcal{H}}\left({({\delta }_{\mathcal{T}}^2+{\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }\right)}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

Therefore,

$$\frac{n_{\mathcal{T}}{m}_{\mathcal{H}}{({\Delta }_{\mathcal{T}}^2+{\Delta }_{\mathcal{H}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}\le Y_1\le \frac{n_{\mathcal{T}}{m}_{\mathcal{H}}\left({({\delta }_{\mathcal{T}}^2+{\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }\right)}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}.$$

(16)

Similarly, one can show that

$$\frac{n_{\mathcal{H}}{m}_{\mathcal{T}}{({\Delta }_{\mathcal{H}}^2+{\Delta }_{\mathcal{T}}^2+2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\delta }_{\mathcal{T}}+{\delta }_{\mathcal{H}})}^{\beta }}\le Y_2\le \frac{n_{\mathcal{H}}{m}_{\mathcal{T}}{({\delta }_{\mathcal{T}}^2+{\delta }_{\mathcal{H}}^2+2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }}{2^{\beta }{({\Delta }_{\mathcal{T}}+{\Delta }_{\mathcal{H}})}^{\beta }}.$$

(17)

By adding the Equations (16) and (17), the desired result is obtained. □

Example 3.

Let$S=P_n\square C_m$and$R=C_n\square C_m$. Then,$S=TU{C}_4(n,m)$is a$C_4$-nanotube (seeFigure 2), and$R=T{C}_4(n,m)$is a$C_4$-nanotorus (seeFigure 2). Now, using Theorem 4, we can obtain the following bounds on$S_{\alpha ,\beta }\left(S\right)$$C_4$-nanotube:

(1). 

If$\alpha ,\beta \ge 0$, then

$$\frac{m}{6^{\beta }}[4^{\alpha }(4n-4+2^{\alpha }(2n-5))+2(1+6^{\alpha }+2^{\alpha })]\le S_{\alpha ,\beta }\left(S\right)\le \frac{m{16}^{\alpha }(2n-1)}{2^{3\beta }},$$

(2). 

If$\alpha \ge 0$and$\beta \le 0$, then

$$\frac{m}{2^{3\beta }}[4^{\alpha }(4n-4+2^{\alpha }(2n-5))+2(1+6^{\alpha }+2^{\alpha })]\le S_{\alpha ,\beta }\left(S\right)\le \frac{m{16}^{\alpha }(2n-1)}{6^{\beta }},$$

(3). 

If$\alpha \le 0$and$\beta \ge 0$, then

$$\frac{m{16}^{\alpha }(2n-1)}{2^{3\beta }}\le S_{\alpha ,\beta }\left(S\right)\le \frac{m{9}^{\alpha }(2n-1)}{6^{\beta }},$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\frac{m{16}^{\alpha }(2n-1)}{6^{\beta }}\le S_{\alpha ,\beta }\left(S\right)\le \frac{m{9}^{\alpha }(2n-1)}{2^{3\beta }}.$$

Similarly, one can obtain the bounds on$S_{\alpha ,\beta }\left(R\right)$for$C_4$-nanotorus.

2.5. Disjunction of Graphs

For the graphs $\mathcal{T}$ and $\mathcal{H},$ the disjunction $\mathcal{T}\ominus \mathcal{H}$ is the graph with vertex set $V\left(\mathcal{T}\right)\times V\left(\mathcal{H}\right)$ and $(w_1,z_1)(w_2,z_2)\in E(\mathcal{T}\ominus \mathcal{H})$ if and only if $w_1{w}_2\in E\left(\mathcal{T}\right)$ or $z_1{z}_2\in E\left(\mathcal{H}\right)$. For any vertex $(w,z)\in V(\mathcal{T}\ominus \mathcal{H})$, we have:

$$d_{\mathcal{T}\ominus \mathcal{H}}\left((w,z)\right)=n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z\right)-d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right).$$

(18)

Theorem 5.

Suppose$\mathcal{T}$and$\mathcal{H}$are two graphs. Then:

(1). 

If$\alpha ,\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(2). 

If$\alpha \ge 0$and$\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}-\frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}-\frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}.\hfill \end{array}$$

(3). 

If$\alpha \le 0$and$\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}-\frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}-\frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}.\hfill \end{array}$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

In all cases, the inequality becomes an equality if$\mathcal{T}$and$\mathcal{H}$are regular graphs.

Proof. 

$\left(1\right).$ Let $\alpha ,\beta \ge 0$. From the definition of generalized ISI index and Equation (18), we obtain:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\ominus \mathcal{H})& =& \sum _{(w_1,z_1)(w_2,z_2)\in E(\mathcal{T}\ominus \mathcal{H})}\left(\frac{{\left(d_{\mathcal{T}\square \mathcal{H}}\left((w_1,z_1)\right)d_{\mathcal{T}\ominus \mathcal{H}}\left((w_2,z_2)\right)\right)}^{\alpha }}{{(d_{\mathcal{T}\ominus \mathcal{H}}\left((w_1,z_1)\right)+d_{\mathcal{T}\ominus \mathcal{H}}\left((w_2,z_2)\right))}^{\beta }}\right)\hfill \\ & =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}^{\left(w_2\right)})+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}^{\left(z_1\right)}-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& +& \sum _{\begin{array}{c}{z}_1\in V\left(\mathcal{H}\right)\\ z_2\in V\left(\mathcal{H}\right)\\ w_1{w}_2\in E\left(\mathcal{T}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}^{\left(w_2\right)})+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}^{\left(z_1\right)}-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}{w}_1{w}_2\in E\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & =& Q_1^{\prime }+Q_2^{\prime }+Q_3^{\prime },\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill Q_1^{\prime }& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}^{\left(w_1\right)}{d}_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}^{\left(w_2\right)})+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}^{\left(z_1\right)}-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right),\hfill \\ \hfill Q_2^{\prime }& =& \sum _{\begin{array}{c}{z}_1\in V\left(\mathcal{H}\right)\\ z_2\in V\left(\mathcal{H}\right)\\ w_1{w}_2\in E\left(\mathcal{T}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}^{\left(w_1\right)}{d}_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}^{\left(w_2\right)})+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}^{\left(z_1\right)}-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right),\hfill \\ \hfill Q_3^{\prime }& =& \sum _{\begin{array}{c}{w}_1{w}_2\in E\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right).\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill Q_1^{\prime }& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \ge & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Q_1^{\prime }& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \le & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{H}}{\delta }_{\mathcal{T}}+2n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{H}}^2{m}_{\mathcal{T}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }},\hfill \end{array}$$

Therefore,

$$\frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\le Q_1^{\prime }\le \frac{n_{\mathcal{H}}^2{m}_{\mathcal{T}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}.$$

(19)

Similarly, one can show that

$$\begin{array}{ccc}\hfill \frac{n_{\mathcal{H}}^2{m}_{\mathcal{T}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}& \le & Q_2^{\prime }\le \frac{n_{\mathcal{H}}^2{m}_{\mathcal{T}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }},\hfill \\ \hfill \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}& \le & Q_3^{\prime }\le \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(20)

By adding the Equations (19) and (20), the desired result is obtained.

$\left(2\right).$ Let $\alpha \ge 0$ and $\beta \le 0$. Then,

$$\begin{array}{ccc}\hfill Q_1^{\prime }& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_2\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}^{\left(z_2\right)})\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \ge & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{H}}{\delta }_{\mathcal{T}}+2n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}=\frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Q_1^{\prime }& =& \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{\left((n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w_1\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right))(n_{\mathcal{H}}{d}_{\mathcal{T}}^{\left(w_2\right)}+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z_2\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))\right)}^{\alpha }}{{(n_{\mathcal{H}}(d_{\mathcal{T}}\left(w_1\right)+d_{\mathcal{T}}\left(w_2\right))+n_{\mathcal{T}}(d_{\mathcal{H}}\left(z_1\right)+d_{\mathcal{H}}\left(z_2\right))-d_{\mathcal{T}}\left(w_1\right)d_{\mathcal{H}}\left(z_1\right)-d_{\mathcal{T}}\left(w_2\right)d_{\mathcal{H}}\left(z_2\right))}^{\beta }}\right)\hfill \\ & \le & \sum _{\begin{array}{c}{w}_1\in V\left(\mathcal{T}\right)\\ w_2\in V\left(\mathcal{T}\right)\\ z_1{z}_2\in E\left(\mathcal{H}\right)\end{array}}\left(\frac{{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\alpha }}{{(2n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+2n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\right)\hfill \\ & =& \frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}=\frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}.\hfill \end{array}$$

Therefore,

$$\frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}\le Q_1^{\prime }\le \frac{n_{\mathcal{T}}^2{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}$$

(21)

Similarly, we can show that

$$\begin{array}{ccc}\hfill \frac{n_{\mathcal{H}}^2{m}_{\mathcal{T}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}& \le & Q_2^{\prime }\le \frac{n_{\mathcal{H}}^2{m}_{\mathcal{T}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }},\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}& \le & Q_3^{\prime }\le \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}.\hfill \end{array}$$

(23)

The desired result is obtained by adding the Equations (21) and (22). Parts $\left(3\right)$ and $\left(4\right)$ can be proven analogously. □

2.6. Symmetric Difference of Graphs

For the graphs $\mathcal{T}$ and $\mathcal{H}$, the symmetric difference $\mathcal{T}\oplus \mathcal{H}$ is the graph with a vertex set $V\left(\mathcal{T}\right)\times V\left(\mathcal{H}\right)$ and $(w_1,z_1)(w_2,z_2)\in E(\mathcal{T}\oplus \mathcal{H})$ if and only if $w_1{w}_2\in E\left(\mathcal{T}\right)$ or $z_1{z}_2\in E\left(\mathcal{H}\right)$, but not both. For any vertex $(w,z)\in V(\mathcal{T}\oplus \mathcal{H})$, we have:

$$d_{\mathcal{T}\oplus \mathcal{H}}\left((w,z)\right)=n_{\mathcal{H}}{d}_{\mathcal{T}}\left(w\right)+n_{\mathcal{T}}{d}_{\mathcal{H}}\left(z\right)-2d_{\mathcal{T}}\left(w\right)d_{\mathcal{H}}\left(z\right).$$

(24)

The proof of the next theorem is similar to the proof of Theorem 5 and is thus neglected.

Theorem 6.

Suppose$\mathcal{T}$and$\mathcal{H}$are two graphs. Then,

(1). 

If $\alpha ,\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{2m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

(2). 

If $\alpha \ge 0$ and $\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}.\hfill \end{array}$$

(3). 

If $\alpha \le 0$ and $\beta \ge 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha -\beta }}{2^{\beta }}.\hfill \end{array}$$

(4). 

If$\alpha ,\beta \le 0$, then

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \ge & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }},\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\oplus \mathcal{H})& \le & \frac{(n_{\mathcal{T}}^2{m}_{\mathcal{H}}+n_{\mathcal{H}}^2{m}_{\mathcal{T}}){(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-2{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-2{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{\beta }}\hfill \\ & -& \frac{4m_{\mathcal{T}}{m}_{\mathcal{H}}{(n_{\mathcal{H}}{\Delta }_{\mathcal{T}}+n_{\mathcal{T}}{\Delta }_{\mathcal{H}}-{\delta }_{\mathcal{T}}{\delta }_{\mathcal{H}})}^{2\alpha }}{2^{\beta }{(n_{\mathcal{H}}{\delta }_{\mathcal{T}}+n_{\mathcal{T}}{\delta }_{\mathcal{H}}-{\Delta }_{\mathcal{T}}{\Delta }_{\mathcal{H}})}^{\beta }}.\hfill \end{array}$$

In all the cases, the inequality becomes an equality if$\mathcal{T}$and$\mathcal{H}$are regular graphs.

3. Exact Formulas of ${\mathbf{S}}_{\mathbf{\alpha },\mathbf{\beta }}\left(\mathcal{T}\right)$ for Some Graph Operations

In this section, we find the exact formulas of the generalized ISI index for the disjoint union and the splicing and linking of graphs.

3.1. Disjoint Union

The disjoint union $\mathcal{T}\cup \mathcal{H}$ of $\mathcal{T}$ and $\mathcal{H}$ is the graph with $V(\mathcal{T}\cup \mathcal{H})=V\left(\mathcal{T}\right)\cup V\left(\mathcal{H}\right)$ and $E(\mathcal{T}\cup \mathcal{H})=E\left(\mathcal{T}\right)\cup E\left(\mathcal{H}\right)$. The proof of the next theorem is simple and is thus omitted.

Theorem 7.

Let${\mathcal{T}}_1,\cdots ,{\mathcal{T}}_s$be the graphs with disjoint vertex sets. Then, the generalized ISI index of${\mathcal{T}}_1\cup {\mathcal{T}}_2\cdots \cup {\mathcal{T}}_s$is given by:

$$S_{\alpha ,\beta }({\mathcal{T}}_1\cup {\mathcal{T}}_2\cdots \cup {\mathcal{T}}_s)=S_{\alpha ,\beta }\left({\mathcal{T}}_1\right)+S_{\alpha ,\beta }\left({\mathcal{T}}_2\right)+\cdots +S_{\alpha ,\beta }\left({\mathcal{T}}_s\right)$$

3.2. Splicing of Graphs

The splice $\mathcal{T}\odot \mathcal{H}$ is a graph formed from graphs $\mathcal{T}$ and $\mathcal{H}$ by identifying a vertex $q\in V\left(\mathcal{T}\right)$ and a vertex $r\in V\left(\mathcal{H}\right)$ in $\mathcal{T}\cup \mathcal{H}$. Let y be the vertex which is obtained by identifying q and r. Now for any $z\in V(\mathcal{T}\odot \mathcal{H})(q,r)$, we have:

$$d_{(\mathcal{T}\odot \mathcal{H})(q,r)}\left(z\right)=\left\{\begin{array}{cc}{d}_{\mathcal{T}}\left(z\right)\hfill & \mathrm{if} z\in V\left(\mathcal{T}\right) \mathrm{and} z\ne q,\hfill \\ d_{\mathcal{H}}\left(z\right)\hfill & \mathrm{if} z\in V\left(\mathcal{H}\right) \mathrm{and} z\ne r,\hfill \\ d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)\hfill & \mathrm{if} z=y.\hfill \end{array}\right.$$

(25)

The next theorem gives exact formulae of $S_{\alpha ,\beta }$ for the splicing of graphs.

Theorem 8.

The generalized ISI index of the splicing of two graphs$\mathcal{T}$and$\mathcal{H}$with respect to vertices$q\in V\left(\mathcal{T}\right)$and$r\in V\left(\mathcal{H}\right)$is given by:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\odot \mathcal{H})(q,r)& =& S_{\alpha ,\beta }\left(\mathcal{T}\right)+S_{\alpha ,\beta }\left(\mathcal{H}\right)+q_1^{\prime }+q_2^{\prime },\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill q_1^{\prime }& =& \sum _{z\in N_{\mathcal{T}}\left(q\right)}\left(\frac{{(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)+d_{\mathcal{H}}\left(r\right)d_{\mathcal{T}}\left(z\right))}^{\alpha }}{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)}+d_{\mathcal{T}}{\left(z\right))}^{\beta }-\frac{{\left(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right),\hfill \\ \hfill q_2^{\prime }& =& \sum _{z\in N_{\mathcal{T}}\left(r\right)}\left(\frac{{(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)+d_{\mathcal{T}}\left(q\right)d_{\mathcal{H}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}-\frac{{\left(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right).\hfill \end{array}$$

Proof. 

$\left(1\right).$ By the definition of generalized ISI index and Equation (25), we obtain:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\odot \mathcal{H})(q,r)& =& \sum _{\begin{array}{c}wz\in E\left(\mathcal{T}\right)\\ w,z\ne q\end{array}}\left(\frac{{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right)+\sum _{\begin{array}{c}wz\in E\left(\mathcal{H}\right)\\ w,z\ne r\end{array}}\left(\frac{{\left(d_{\mathcal{H}}\left(w\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{H}}\left(w\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}wz\in E\left(\mathcal{T}\right),w=q\\ z\in V\left(\mathcal{T}\right)\end{array}}\left(\frac{{(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)+d_{\mathcal{H}}\left(r\right)d_{\mathcal{T}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{H}}\left(r\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}wz\in E\left(\mathcal{H}\right),w=r\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{(d_{\mathcal{H}}\left(w\right)d_{\mathcal{H}}\left(z\right)+d_{\mathcal{T}}\left(q\right)d_{\mathcal{H}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(w\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ & =& S_{\alpha ,\beta }\left(\mathcal{T}\right)-\sum _{z\in N_{\mathcal{T}}\left(q\right)}\left(\frac{{\left(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right)+S_{\alpha ,\beta }\left(\mathcal{H}\right)\hfill \\ & -& \sum _{z\in N_{\mathcal{T}}\left(r\right)}\left(\frac{{\left(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ & +& \sum _{z\in N_{\mathcal{T}}\left(q\right)}\left(\frac{{(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)+d_{\mathcal{H}}\left(r\right)d_{\mathcal{T}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right)\hfill \\ & +& \sum _{z\in N_{\mathcal{H}}\left(r\right)}\left(\frac{{(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)+d_{\mathcal{T}}\left(q\right)d_{\mathcal{H}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ \hfill S_{\alpha ,\beta }(\mathcal{T}\odot \mathcal{H})(q,r)& =& S_{\alpha ,\beta }\left(\mathcal{T}\right)+S_{\alpha ,\beta }\left(\mathcal{H}\right)+q_1^{\prime }+q_2^{\prime }.\hfill \end{array}$$

This proves the result. □

3.3. Linking of Graphs

The link $\mathcal{T}\sim \mathcal{H}$ of graphs $\mathcal{T}$ and $\mathcal{H}$ can be formed from $\mathcal{T}$ and $\mathcal{H}$ by making a vertex $q\in V\left(\mathcal{T}\right)$ and a vertex $r\in V\left(\mathcal{H}\right)$ adjacent in $\mathcal{T}\cup \mathcal{H}$. Now, for any $z\in V(\mathcal{T}\sim \mathcal{H})(q,r)$, we have:

$$d_{(\mathcal{T}\sim \mathcal{H})(q,r)}\left(z\right)=\left\{\begin{array}{cc}{d}_{\mathcal{T}}\left(z\right)\hfill & \mathrm{if} z\in V\left(\mathcal{T}\right) \mathrm{and} z\ne q,\hfill \\ d_{\mathcal{H}}\left(z\right)\hfill & \mathrm{if} z\in V\left(\mathcal{H}\right) \mathrm{and} z\ne r,\hfill \\ d_{\mathcal{T}}\left(z\right)+1\hfill & \mathrm{if} z=q,\hfill \\ d_{\mathcal{H}}\left(z\right)+1\hfill & \mathrm{if} z=r,\hfill \end{array}\right.$$

(26)

The following theorem gives exact formulae of $S_{\alpha ,\beta }$ for the linking of graphs.

Theorem 9.

The generalized ISI index of the linking of two graphs$\mathcal{T}$and H with respect to vertices$q\in V\left(\mathcal{T}\right)$and$r\in V\left(\mathcal{H}\right)$is given by:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\sim \mathcal{H})(q,r)& =& S_{\alpha ,\beta }\left(\mathcal{T}\right)+S_{\alpha ,\beta }\left(\mathcal{H}\right)+\left(\frac{{(d_{\mathcal{T}}\left(q\right)+1)}^{\alpha }{(d_{\mathcal{H}}^{\left(r\right)}+1)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+2)}^{\beta }}\right)+k_1+k_2,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill k_1& =& \sum _{z\in N_{\mathcal{T}}\left(q\right)}\left(\frac{({(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)+d_{\mathcal{T}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{T}}\left(z\right)+1)}^{\beta }}-\frac{{\left(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right),\hfill \end{array}$$

$$\begin{array}{ccc}\hfill k_2& =& \sum _{z\in N_{\mathcal{T}}\left(r\right)}\left(\frac{({(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)+d_{\mathcal{H}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right)+1)}^{\beta }}-\frac{{\left(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right).\hfill \end{array}$$

Proof. 

$\left(1\right).$ By the definition of generalized ISI index and Equation (26), we have:

$$\begin{array}{ccc}\hfill S_{\alpha ,\beta }(\mathcal{T}\sim \mathcal{H})(q,r)& =& \sum _{\begin{array}{c}wz\in E\left(\mathcal{T}\right)\\ w,z\ne q\end{array}}\left(\frac{{\left(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}^{\left(w\right)}+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right)+\sum _{\begin{array}{c}wz\in E\left(\mathcal{H}\right)\\ w,z\ne r\end{array}}\left(\frac{{\left(d_{\mathcal{H}}\left(w\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{H}}\left(w\right)+d_{\mathcal{H}}\left(z\right))}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}wz\in E\left(\mathcal{T}\right),w=q\\ z\in V\left(\mathcal{T}\right)\end{array}}\left(\frac{{(d_{\mathcal{T}}\left(w\right)d_{\mathcal{T}}\left(z\right)+d_{\mathcal{T}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(w\right)+d_{\mathcal{T}}\left(z\right)+1)}^{\beta }}\right)\hfill \\ & +& \left(\frac{{(d_{\mathcal{T}}\left(q\right)+1)}^{\alpha }{(d_{\mathcal{H}}\left(r\right)+1)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+2)}^{\beta }}\right)\hfill \\ & +& \sum _{\begin{array}{c}wz\in E\left(\mathcal{H}\right),w=r\\ z\in V\left(\mathcal{H}\right)\end{array}}\left(\frac{{(d_{\mathcal{H}}\left(w\right)d_{\mathcal{H}}\left(z\right)+d_{\mathcal{H}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{H}}\left(w\right)+d_{\mathcal{H}}\left(z\right)+1)}^{\beta }}\right)\hfill \\ & =& S_{\alpha ,\beta }\left(\mathcal{T}\right)-\sum _{z\in N_{\mathcal{T}}\left(q\right)}\left(\frac{{\left(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{T}}\left(z\right))}^{\beta }}\right)+S_{\alpha ,\beta }\left(\mathcal{H}\right)\hfill \\ & -& \sum _{z\in N_{\mathcal{T}}\left(r\right)}\left(\frac{{\left(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)\right)}^{\alpha }}{{(d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(v\right))}^{\beta }}\right)\hfill \\ & +& \sum _{z\in N_{\mathcal{T}}\left(q\right)}\left(\frac{{(d_{\mathcal{T}}\left(q\right)d_{\mathcal{T}}\left(z\right)+d_{\mathcal{T}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{T}}\left(z\right)+1)}^{\beta }}\right)+\left(\frac{{(d_{\mathcal{T}}\left(q\right)+1)}^{\alpha }{(d_{\mathcal{H}}\left(r\right)+1)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+2)}^{\beta }}\right)\hfill \\ & +& \sum _{z\in N_{\mathcal{H}}\left(r\right)}\left(\frac{{(d_{\mathcal{H}}\left(r\right)d_{\mathcal{H}}\left(z\right)+d_{\mathcal{H}}\left(z\right))}^{\alpha }}{{(d_{\mathcal{H}}\left(r\right)+d_{\mathcal{H}}\left(z\right)+1)}^{\beta }}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& =& S_{\alpha ,\beta }\left(\mathcal{T}\right)+S_{\alpha ,\beta }\left(\mathcal{H}\right)+\left(\frac{{(d_{\mathcal{T}}\left(q\right)+1)}^{\alpha }{(d_{\mathcal{H}}\left(r\right)+1)}^{\alpha }}{{(d_{\mathcal{T}}\left(q\right)+d_{\mathcal{H}}\left(r\right)+2)}^{\beta }}\right)+k_1+k_2.\hfill \end{array}$$

Hence, we have the desired result. □

4. Conclusions

It is noteworthy to highlight that as a considerable number of graphs are formed by smaller ones using graph products (and, as a result, their properties are highly related), the analysis of graph products is a convenient and reasonable research subject. A new graph can be made from the provided graphs by employing graph operations, and it is shown that many chemical graphs can be formed through graph operations. The relations gained for distinct topological indices of graph operations are in the form of topological invariants of their components; thus, it is useful to compute the topological descriptors of some nanostructures and molecular graphs. This paper finds bounds on the generalized ISI index for Kronecker products, joining, corona products, Cartesian products, disjunction, and symmetric differences of graphs. We also obtain exact formulas of the generalized ISI index for the disjoint union, linking, and splicing of graphs. In the future, we will work on some extremal results for the generalized ISI index of trees, unicyclic graphs, bicyclic graphs, etc., with some fixed parameters such as fixed pendent vertices, fixed branches, degree sequence, maximum and minimum degrees, etc.