On the Extended Adjacency Eigenvalues of a Graph

Abstract

Let $\mathcal{H}$ be a graph of order n with m edges. Let $d_i=d\left(v_i\right)$ be the degree of the vertex $v_i$. The extended adjacency matrix $A_{ex}\left(\mathcal{H}\right)$ of $\mathcal{H}$ is an $n\times n$ matrix defined as $A_{ex}\left(\mathcal{H}\right)=\left(b_{ij}\right)$, where $b_{ij}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)$, whenever $v_i$ and $v_j$ are adjacent and equal to zero otherwise. The largest eigenvalue of $A_{ex}\left(\mathcal{H}\right)$ is called the extended adjacency spectral radius of $\mathcal{H}$ and the sum of the absolute values of its eigenvalues is called the extended adjacency energy of $\mathcal{H}$. In this paper, we obtain some sharp upper and lower bounds for the extended adjacency spectral radius in terms of different graph parameters and characterize the extremal graphs attaining these bounds. We also obtain some new bounds for the extended adjacency energy of a graph and characterize the extremal graphs attaining these bounds. In both cases, we show our bounds are better than some already known bounds in the literature.

1. Introduction

Let $\mathcal{H}=\left(V\right(\mathcal{H}),E(\mathcal{H}\left)\right)$ be a graph, where $V\left(\mathcal{H}\right)=\{v_1,v_2,\dots ,v_n\}$ is the set of vertices and $E\left(\mathcal{H}\right)$ is the set of edges in $\mathcal{H}$. A graph $\mathcal{H}$ is called a simple graph if there are no loops or multiple edges. Two vertices u and v are said to be adjacent if there is an edge between them and non-adjacent otherwise. A graph $\mathcal{H}$ is called connected if any two vertices $v_i,v_j\in V\left(\mathcal{H}\right)$ can be connected by paths from $v_i$ to $v_j$. Throughout this paper, we confine ourselves to connected simple graphs. For any notions not defined explicitly in the paper, we refer the reader to the standard book [1].

The degree of the vertex $v_i\in V\left(\mathcal{H}\right)$ is the number of edges adjacent to $v_i$ and it is denoted by $d_i=d\left(v_i\right)$. The adjacency matrix of a graph D is $A\left(\mathcal{H}\right)=\left(a_{ij}\right)$, where $a_{ij}=1$ if $v_i$ and $v_j$ are adjacent and $a_{ij}=0$ otherwise. The eigenvalues of $A\left(\mathcal{H}\right)$ are called the eigenvalues of $\mathcal{H}$ and its largest eigenvalue is the spectral radius of $\mathcal{H}$. The energy of $\mathcal{H}$, denoted by $\mathcal{E}\left(\mathcal{H}\right)$, is defined as the sum of the absolute values of the eigenvalues of $A\left(\mathcal{H}\right)$. That is, $\mathcal{E}\left(\mathcal{H}\right)={\displaystyle \sum _{i=1}^n}\left|{\lambda }_i\right|$, where ${\lambda }_i$, $1\le i\le n$ are the eigenvalues of $A\left(\mathcal{H}\right)$. The spectral properties, such as the bounds for the spectral radius and the energy of graphs, are well studied and many papers can be found in the literature on this theme. For some recent papers in this direction and related results, we refer to [2,3,4,5].

An extended version of the adjacent matrix of a graph $\mathcal{H}$ was considered in [6] by defining the extended adjacency matrix $A_{ex}\left(\mathcal{H}\right)$ as a square matrix of order n having $(i,j)$-entry is equal to ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)$, if $v_i$ adjacent to $v_j$ and zero otherwise. The eigenvalues of $A_{ex}\left(\mathcal{H}\right)$ are called the extended adjacency eigenvalues of $\mathcal{H}$. The largest eigenvalue of $A_{ex}\left(\mathcal{H}\right)$ is the extended adjacency spectral radius of $\mathcal{H}$ and it is denoted by $\lambda \left(A_{ex}\right)$. The sum of the absolute values of the extended adjacency eigenvalues is the extended adjacency energy ${\mathcal{E}}_{ex}\left(\mathcal{H}\right)$ of $\mathcal{H}$. That is,

$${\mathcal{E}}_{ex}\left(\mathcal{H}\right)=\sum _{i=1}^n\left|{\eta }_i\right|,$$

where ${\eta }_1,{\eta }_2,\cdots ,{\eta }_n$ are the extended adjacency eigenvalues of $\mathcal{H}$.

In the research presented in [6], the concept of extended adjacency matrices was utilized to explore the effects of heteroatoms and multiple bonds on molecular structures. This investigation revealed that both the extended adjacency spectral radius and the extended adjacency energy demonstrate significant discriminative abilities, correlating effectively with various physicochemical properties and biological activities of organic compounds. Furthermore, recent studies by Gutman et al. [7] delved into how the extended adjacency energy is influenced by molecular structure, outlining its fundamental properties. They discovered a linear relationship between the extended adjacency energy and the total $\pi$-electron energy, specifically noting that the difference between these two quantities is directly proportional to the difference between the number of edges and the GA index of the molecule.

The scope of the extended adjacency spectral radius and energy has been broadened in subsequent research, yielding intriguing insights as documented in [8,9,10,11,12,13,14,15]. These studies collectively contribute to a deeper understanding of the spectral characteristics and their implications in molecular chemistry. In this paper, we aim to investigate the extended adjacency spectral radius and energy of graphs further. Our objective is to establish new bounds for these spectral graph invariants by leveraging novel graph parameters. By doing so, we hope to provide a more comprehensive framework for analyzing the spectral properties of graphs, thereby enhancing the predictive power and applicability of these mathematical tools in various scientific domains.

The structure of the remaining sections of this paper is organized to systematically address the main research objectives. In Section 2, we derive precise upper and lower bounds for the extended adjacency spectral radius of a graph. This section includes a detailed characterization of the extremal graphs that achieve these bounds, demonstrating the superiority of our proposed bounds over several existing ones in the literature. We provide rigorous proofs and comparative analyses to highlight the improvements. Section 3 builds upon the findings from Section 2. We apply the previously obtained results to establish new bounds for the extended adjacency energy of a graph. This section also involves a thorough characterization of the extremal graphs that meet these new bounds. Through comprehensive evaluations, we illustrate that our bounds surpass some of the known bounds, thereby contributing significantly to the field.

2. Extended Adjacency Spectral Radius

In this section, we derive novel upper and lower bounds for the extended adjacency spectral radius of a graph. These bounds are expressed in terms of various structural parameters intrinsic to the graph’s configuration. By thoroughly examining these parameters, we identify and characterize the extremal graphs that achieve these newly established bounds. Our approach not only refines existing methodologies but also introduces new perspectives on understanding the spectral properties of graphs through these bounds.

To set the stage for our findings, we first present a significant result that establishes a relationship between the spectral radii of non-negative matrices. This result, which serves as a foundational tool in our analysis, can be traced back to the work documented in [16]. By leveraging this relationship, we can draw insightful connections between the structural aspects of graphs and their spectral characteristics.

Lemma 1. 

Let C and D be non-negative matrices with their spectral radii $\lambda \left(C\right)$ and $\lambda \left(D\right)$, respectively. If $0\le C\le D$, then $\lambda \left(C\right)\le \lambda \left(D\right)$. Furthermore, if D is irreducible and $0\le C<D$, then $\lambda \left(C\right)<\lambda \left(D\right)$.

The following relation between the extended adjacency spectral radius and the spectral radius of a graph $\mathcal{H}$ was established in [11]:

$$\begin{array}{c}\hfill \lambda \left(A\left(\mathcal{H}\right)\right)\le \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)\lambda \left(A\left(\mathcal{H}\right)\right),\end{array}$$

(1)

where $\lambda \left(A\right(\mathcal{H}\left)\right)$ is the spectral radius of $\mathcal{H}$. Equality occurs on the left if, and only if, $\mathcal{H}$ is a regular graph and occurs on the right if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

Let $\mathcal{H}$ be a graph that has the property that no two of its vertices of the same degree are adjacent. Then, we can improve the lower bound given in (1) as follows:

Theorem 1. 

Let $\mathcal{H}$ be a graph of order n. Suppose that no two vertices of the same degree are adjacent in $\mathcal{H}$. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge (1+\epsilon )\lambda \left(A\left(\mathcal{H}\right)\right),\end{array}$$

where $\epsilon >0$ is a constant given by $\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\left\{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\right\}=1+\epsilon$. Equality occurs if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

Proof. 

Since no two vertices of the same degree are adjacent in $\mathcal{H}$, it follows that ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge 1+\epsilon$, where $\epsilon >0$. Equality occurs if, and only if, ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)=1+\epsilon$, that is, if, and only if, ${\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}=2+2\epsilon$. Now, ${\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}$ is a constant for all $v_i$ adjacent to $v_j$ gives that $d_i$ is a constant and $d_j$ is a constant different from $d_i$. This, together with the fact that no two vertices of the same degree are adjacent in $\mathcal{H}$, gives that equality occurs if, and only if, $\mathcal{H}$ is a bipartite graph with partite sets $V_1$ and $V_2$, such that the degree of each vertex in $V_1$ is a constant $\Delta$ and the degree of each vertex in $V_2$ is a constant different from $\Delta$. Now, ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge 1+\epsilon$ gives that $A_{ex}\left(\mathcal{H}\right)\ge (1+\epsilon )A\left(\mathcal{H}\right)$ and so by Lemma 1, we obtain $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge (1+\epsilon )\lambda \left(A\left(\mathcal{H}\right)\right)$. This completes the proof. □

Combining the lower bound given in (1) with Theorem 1, we arrive at the following result:

Corollary 1. 

Let $\mathcal{H}$ be a graph of order n. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge (1+\epsilon )\lambda \left(A\left(\mathcal{H}\right)\right),\end{array}$$

where $\epsilon \ge 0$ is a constant given by $\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\left\{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\right\}=1+\epsilon$. For $\epsilon =0,$ the equality occurs if, and only if, $\mathcal{H}$ is a regular graph, and for $\epsilon >0$, the equality occurs if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

The value of the positive constant $\epsilon$ can be arbitrary. For example, consider the double star $D_n(a,b)$, with $a+b=n-2$, $n\ge 4$. For $D_n(a,b)$, we have ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)={\displaystyle \frac{a+1}{2}}\mathrm{or}{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)={\displaystyle \frac{b+1}{2}}\mathrm{or}{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{a}{b}}+{\displaystyle \frac{b}{a}}\right)$. Moreover, for $a<b$, no two vertices of the same degree are adjacent. If $a=5$ and $b=10$, then ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge {\displaystyle \frac{5}{4}}=1+0.5$, giving that $\epsilon =0.5$; if $a=4$ and $b=10$, then ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge {\displaystyle \frac{29}{20}}=1+0.405$; if $a=5$ and $b=20$, then ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge {\displaystyle \frac{17}{8}}=1+1.125$; and if $a=5$ and $b=50$, then ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge 3=1+2$.

The Sombor matrix $S\left(\mathcal{H}\right)=\left(s_{ij}\right)$ (see [5]) of a graph $\mathcal{H}$ is a square matrix of order $n,$ where $s_{ij}=\sqrt{d_{v_i}^2+d_{v_j}^2}$, whenever $v_i$ is adjacent to $v_j$, and equal to $0,$ otherwise. The eigenvalues of $S\left(\mathcal{H}\right)$ are called Sombor eigenvalues of $\mathcal{H}$ and its largest eigenvalue is the Sombor spectral radius of $\mathcal{H}$. For some recent works on the Sombor matrix, we refer to [5] and the references therein. Next, we obtain a relation between the extended adjacency spectral radius and the Sombor spectral radius of a graph $\mathcal{H}$.

Theorem 2. 

Let $\mathcal{H}$ be a graph of order n. Let ${\Delta }_1,{\Delta }_2,{\delta }_1$ and ${\delta }_2$ be, respectively, the maximum degree, the second maximum degree, the minimum degree and the second minimum degree of $\mathcal{H}$. Then,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{1}{{\Delta }_1^2}}+{\displaystyle \frac{1}{{\Delta }_2^2}}}\lambda \left(S\left(\mathcal{H}\right)\right)\le \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{1}{{\delta }_1^2}}+{\displaystyle \frac{1}{{\delta }_2^2}}}\lambda \left(S\left(\mathcal{H}\right)\right),\end{array}$$

where $\lambda \left(S\right(\mathcal{H}\left)\right)$ is the Sombor spectral radius of $\mathcal{H}$. Equality occurs on the left and the right if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

Proof. 

We have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)={\displaystyle \frac{d_i^2+d_j^2}{2d_i{d}_j}}={\displaystyle \frac{\sqrt{d_i^2+d_j^2}}{2d_i{d}_j}}\sqrt{d_i^2+d_j^2}\end{array}$$

and ${\displaystyle \frac{\sqrt{d_i^2+d_j^2}}{2d_i{d}_j}}={\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{d_i^2}}+{\displaystyle \frac{1}{d_j^2}}}\right)\le {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\delta }_1^2}}+{\displaystyle \frac{1}{{\delta }_2^2}}}\right)$, with equality if, and only if, $d_i={\delta }_1$ and $d_j={\delta }_2$ or $d_i={\delta }_2$ and $d_j={\delta }_1$, whenever there is an edge between $v_i$ and $v_j$. Also, ${\displaystyle \frac{\sqrt{d_i^2+d_j^2}}{2d_i{d}_j}}={\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{d_i^2}}+{\displaystyle \frac{1}{d_j^2}}}\right)\ge {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\Delta }_1^2}}+{\displaystyle \frac{1}{{\Delta }_2^2}}}\right)$, with equality if, and only if, $d_i={\Delta }_1$ and $d_j={\Delta }_2$ or $d_i={\Delta }_2$ and $d_j={\Delta }_1$, whenever there is an edge between $v_i$ and $v_j$. It follows that equality occurs in each case if, and only if, $\mathcal{H}$ is a bipartite graph with partite sets $V_1$ and $V_2$, such that the degree of each vertex in $V_1$ is the same and the degree of each vertex in $V_2$ is the same. Now, ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\ge {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\Delta }_1^2}}+{\displaystyle \frac{1}{{\Delta }_2^2}}}\right)\sqrt{d_i^2+d_j^2}$ gives that $A_{ex}\left(\mathcal{H}\right)\ge {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\Delta }_1^2}}+{\displaystyle \frac{1}{{\Delta }_2^2}}}\right)S\left(\mathcal{H}\right)$ and so by Lemma 1, we obtain $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\Delta }_1^2}}+{\displaystyle \frac{1}{{\Delta }_2^2}}}\right)\lambda \left(S\left(\mathcal{H}\right)\right)$. Again, ${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\le {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\delta }_1^2}}+{\displaystyle \frac{1}{{\delta }_2^2}}}\right)\sqrt{d_i^2+d_j^2}$ gives that $A_{ex}\left(\mathcal{H}\right)\le {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\delta }_1^2}}+{\displaystyle \frac{1}{{\delta }_2^2}}}\right)S\left(\mathcal{H}\right)$ and so by Lemma 1, we obtain $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\left(\sqrt{{\displaystyle \frac{1}{{\delta }_1^2}}+{\displaystyle \frac{1}{{\delta }_2^2}}}\right)\lambda \left(S\left(\mathcal{H}\right)\right)$. Equality cases follow from the above discussion and Lemma 1. This completes the proof. □

In particular, we have the following consequence of Theorem 2:

Corollary 2. 

Consider a graph $\mathcal{H}$ that has n vertices. Let Δ represent the maximum degree of $\mathcal{H}$, and δ denote the minimum degree of $\mathcal{H}$. We can establish the following inequalities for the extended adjacency spectral radius $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$:

$${\displaystyle \frac{1}{\sqrt{2}\Delta }}\lambda \left(S\left(\mathcal{H}\right)\right)\le \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{\sqrt{2}\delta }}\lambda \left(S\left(\mathcal{H}\right)\right),$$

where $\lambda \left(S\right(\mathcal{H}\left)\right)$ denotes the Sombor spectral radius of $\mathcal{H}$. These bounds indicate that the extended adjacency spectral radius is squeezed between two expressions involving the Sombor spectral radius scaled by factors that depend on the maximum and minimum degrees of the graph. Notably, both inequalities turn into equalities precisely when $\mathcal{H}$ is a regular graph, implying that all vertices have the same degree.

The Sombor index $SO\left(\mathcal{H}\right)$ of a graph $\mathcal{H}$ is defined as $SO\left(\mathcal{H}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{H}\right)}}\sqrt{d_u^2+d_v^2}$. For recent developments on the Sombor index, we refer to [17,18] and the references therein. The following lower bound for the Sombor spectral radius in terms of the Sombor index can be obtained by employing the same technique as used in Theorem 7 for the Sombor matrix.

$$\begin{array}{c}\hfill \lambda \left(S\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{SO\left(\mathcal{H}\right)}{n}},\end{array}$$

(2)

with equality if, and only if, ${\displaystyle \sum _{v_i{v}_j\in E\left(\mathcal{H}\right)}}\sqrt{d_i^2+d_j^2}$ is the same for all $i=1,2,\cdots ,n$. In particular, if $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph, then the equality occurs in (2).

Using Theorem 2 together with the lower bound given in (2), we obtain the following lower bound for the extended adjacency spectral radius in terms of the Sombor index of $\mathcal{H}$.

Corollary 3. 

Let $\mathcal{H}$ be a graph of order n. Let Δ and δ be the maximum degree and the minimum degree of $\mathcal{H}$, respectively. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge \sqrt{{\displaystyle \frac{1}{{\Delta }_1^2}}+{\displaystyle \frac{1}{{\Delta }_2^2}}}{\displaystyle \frac{SO\left(\mathcal{H}\right)}{n}},\end{array}$$

with equality if, and only if, $\mathcal{H}$ is a regular graph.

Let $m_i={\displaystyle \frac{1}{d_i}}{\displaystyle \sum _{v_i{v}_j\in E\left(\mathcal{H}\right)}}{d}_j$ and $h_i=d_i{\displaystyle \sum _{v_i{v}_j\in E\left(\mathcal{H}\right)}}{\displaystyle \frac{1}{d_j}}$ be, respectively, the average 2-degree and the harmonic average 2-degree of the vertex $v_i$ in $\mathcal{H}$. The following result gives an upper bound for $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$, in terms of $m_i$ and $h_i$:

Theorem 3. 

Let $\mathcal{H}$ be a connected graph of order n. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{max}\left\{\sqrt{(m_i+h_i)(m_j+h_j)}\right\},\end{array}$$

(3)

with equality if, and only if, $m_i+h_i$ is the same for each vertex $v_i$ in $\mathcal{H}$ or $\mathcal{H}$ is a bipartite graph with partite sets $V_1$ and $V_2$ such that $m_i+h_i$ is the same, for all $v_i\in V_1$ and $m_j+h_j$ is the same, for all $v_j\in V_2$.

Proof. 

Let $\mathcal{H}$ be a connected graph of order n and let $E\left(\mathcal{H}\right)$ be the edge set of $\mathcal{H}$. Let $Z={(z_1,z_2,\dots ,z_n)}^T$ be an eigenvector of $A_{ex}\left(\mathcal{H}\right)$ corresponding to the eigenvalue $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$. We assume that $z_i=max\{z_k:v_k\in V\left(\mathcal{H}\right)\}$ and $z_j=max\{z_k:v_i{v}_k\in E\left(\mathcal{H}\right)\}$. From the i-th equation of $A_{ex}\left(\mathcal{H}\right)Z=\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)Z$, we have

$$\begin{array}{c}\hfill \lambda (A_{ex}\left(\mathcal{H}\right))z_i={\displaystyle \frac{1}{2}}\sum _{v_i{v}_k\in E\left(\mathcal{H}\right)}({\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}})z_k,\end{array}$$

which implies that

$$\begin{array}{cc}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)z_i\le & {\displaystyle \frac{1}{2}}\sum _{v_i{v}_k\in E\left(\mathcal{H}\right)}({\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}})z_j\\ & \le {\displaystyle \frac{1}{2}}(m_i+h_i)z_j.\end{array}$$

(4)

Also, from the j-th equation of $A_{ex}\left(\mathcal{H}\right)Z=\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)Z$, we have

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)z_j={\displaystyle \frac{1}{2}}\sum _{v_j{v}_k\in E\left(\mathcal{H}\right)}({\displaystyle \frac{d_j}{d_k}}+{\displaystyle \frac{d_k}{d_j}})z_k,\end{array}$$

which implies that

$$\begin{array}{cc}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)z_j\le & {\displaystyle \frac{1}{2}}\sum _{v_j{v}_k\in E\left(\mathcal{H}\right)}({\displaystyle \frac{d_j}{d_k}}+{\displaystyle \frac{d_k}{d_j}})z_i\\ & \le {\displaystyle \frac{1}{2}}(m_j+h_j)z_i.\end{array}$$

(5)

By multiplying the corresponding sides of the inequalities (4), (5) and using the fact that $z_k>0$ for all k, we obtain

$$\begin{array}{c}\hfill \lambda (A_{ex}{\left(\mathcal{H}\right)}^2)\le {\displaystyle \frac{1}{4}}(m_i+h_i)(m_j+h_j).\end{array}$$

Thus, the inequality (3) follows as a direct consequence. Now, let us assume that the equality holds in (3). For this to happen, it is necessary that all the intermediate inequalities used in the preceding argument must also hold as equalities. From the equality condition in (5), we deduce that $z_k=z_j$ for every k such that $v_i{v}_k\in E\left(\mathcal{H}\right)$. Similarly, from the equality condition in (6), we find that $z_k=z_i$ for all k such that $v_j{v}_k\in E\left(\mathcal{H}\right)$. Consider the sets $U_1=\{v_k:z_k=z_i\}$ and $U_2=\{v_k:z_k=z_j\}$. It is evident that $N\left(v_i\right)\subseteq U_2$ and $N\left(v_j\right)\subseteq U_1$. We will now demonstrate that $V\left(\mathcal{H}\right)=U_1\cup U_2$. Let $v_p\in N\left(v_i\right)$ and suppose $v_r\in N\left(v_p\right)$; then it follows that $z_p=z_j$ and $z_r=z_i$. Furthermore, if $s\in N\left(v_r\right)$ and $v_r\in N\left(v_p\right)$, then we obtain $z_r=z_j$. Continuing this process, and given that $\mathcal{H}$ is a connected graph, we conclude that for every vertex $u\in V\left(\mathcal{H}\right)$, it holds that $z_u=z_i$ or $z_u=z_j$. This establishes that $V\left(\mathcal{H}\right)=U_1\cup U_2$.

First, we assume that $\mathcal{H}$ is a connected, non-bipartite graph. In this scenario, $\mathcal{H}$ contains odd cycles, and by following the above procedure, we find that $z_i=z_j$. Consequently, $Z=(1,1,\cdots ,1)$ becomes an eigenvector corresponding to $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$, which implies that $\mathcal{H}$ has uniform row sums. Therefore, equality in this case occurs if $m_i+h_i$ is identical for all vertices $v_i\in V\left(\mathcal{H}\right)$. On the other hand, if $\mathcal{H}$ is a connected bipartite graph, we encounter two possibilities: either $z_i=z_j$ or $z_i\ne z_j$. In the event that $z_i=z_j$, as previously discussed, equality is achieved if $m_i+h_i$ remains consistent across all vertices $v_i\in V\left(\mathcal{H}\right)$. Assume that $z_i\ne z_j$. Since, for $v_k\in V_1$, we have $z_k=z_i$ and $z_p=z_j$, for all $v_p\in N\left(v_k\right)$, therefore, it follows that

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)z_j={\displaystyle \frac{1}{2}}(m_j+h_j)z_i.\end{array}$$

This gives that $m_j+h_j={\displaystyle \frac{2\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)z_j}{z_i}}$, for all $v_k\in V_1$. Similarly, for $v_k\in V_2$, we have $z_k=z_j$ and $z_p=z_i$, for all $v_z\in N\left(v_k\right)$, it follows that $m_i+h_i={\displaystyle \frac{2\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)z_i}{z_j}}$, for all $v_k\in V_2$. This shows that equality occurs in this case if $m_i+h_i$ is the same for all $v_i\in V_1$ and $m_i+h_i$ is the same for all $v_i\in V_2$. That is, equality occurs in this case if $m_i+h_i$ is the same for any two vertices belonging to the same partite set.

Conversely, consider the connected graph $\mathcal{H}$ such that $m_i+h_i$ is the same for all $v_i\in V\left(\mathcal{H}\right)$ or $\mathcal{H}$ is a bipartite connected graph with $m_i+h_i$ the same for all the vertices belonging to the same partite set, then it can be seen that equality holds in (3). This completes the proof. □

Since

$$\begin{array}{c}\hfill m_i+h_i=\sum _{v_i{v}_k\in E\left(\mathcal{H}\right)}({\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}})\le {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)d_i,\end{array}$$

we have the following consequence of Theorem 3, which gives an upper bound for the extended adjacency spectral radius in terms of vertex degrees of $\mathcal{H}$.

Theorem 4. 

Let $\mathcal{H}$ be a connected graph of order n. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{max}\left\{\sqrt{d_i{d}_j}\right\},\end{array}$$

(6)

with equality if, and only if, $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph.

Proof. 

The proof of inequality (6) follows from Theorem 3 by using the fact ${\displaystyle \sum _{v_i{v}_k\in E\left(\mathcal{H}\right)}}({\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}})\le {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)d_i$, in the inequalities (4) and (5). Equality occurs if, and only if, equality occurs in Theorem 3 and ${\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}}={\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}$. Since, the equality ${\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}}={\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}$ holds if, and only if, $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph and for regular graphs or for bipartite semi-regular graphs the quantity $m_i+h_i$ is the same for all i. It follows that equality holds in (6) if, and only if, $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph. This completes the proof. □

The following gives a lower bound for $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$, in terms of $m_i$ and $h_i$.

Theorem 5. 

Let $\mathcal{H}$ be a connected graph of order n. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{1}{2}}\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\left\{\sqrt{(m_i+h_i)(m_j+h_j)}\right\},\end{array}$$

(7)

with equality if, and only if, $m_i+h_i$ is the same for each vertex $v_i$ in $\mathcal{H}$ or $\mathcal{H}$ is a bipartite graph with partite sets $V_1$ and $V_2$ such that $m_i+h_i$ is the same, for all $v_i\in V_1$ and $m_j+h_j$ is the same, for all $v_j\in V_2$.

Proof. 

Let $\mathcal{H}$ be a connected graph of order n and let $E\left(\mathcal{H}\right)$ be the edge set of $\mathcal{H}$. Let $Z={(z_1,z_2,\dots ,z_n)}^T$ be an eigenvector of $A_{ex}\left(\mathcal{H}\right)$ corresponding to the eigenvalue $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$. We assume that $z_i=min\{z_k;v_k\in V\left(\mathcal{H}\right)\}$ and $z_j=min\{z_k;v_i{v}_k\in E\left(\mathcal{H}\right)\}$. Now, proceeding with the same technique as employed in the proof of Theorem 3, the result follows. □

The following observation follows from Theorem 5 and gives a lower bound for the extended adjacency spectral radius in terms of the vertex degrees of $\mathcal{H}$.

Theorem 6. 

Let $\mathcal{H}$ be a connected graph of order n. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge (1+\epsilon )\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\left\{\sqrt{d_i{d}_j}\right\}.\end{array}$$

(8)

For $\epsilon =0$, the equality occurs if, and only if, $\mathcal{H}$ is a regular graph, and for $\epsilon >0$, the equality occurs if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

Proof. 

The proof of the inequality (8) follows from Theorem 5 by using the fact that ${\displaystyle \sum _{v_i{v}_k\in E\left(\mathcal{H}\right)}}({\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}})\ge (1+\epsilon )d_i$, where $\epsilon$ is a non-negative arbitrary constant fixed for a given graph. Equality holds in (8) if, and only if, the equality holds in Theorem 5 and ${\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}}=1+\epsilon$. If $\epsilon =0$, then ${\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}}=1$, given that $\mathcal{H}$ is a regular graph. If $\epsilon >0$, then ${\displaystyle \frac{d_i}{d_k}}+{\displaystyle \frac{d_k}{d_i}}=1+\epsilon$ gives by using Theorem 1 that $\mathcal{H}$ is a bipartite semi-regular graph. Since, for regular graphs or for bipartite semi-regular graphs, the quantity $m_i+h_i$ is the same for all i, it follows that equality holds in (8) if, and only if, $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph. This completes the proof. □

Remark 1. 

For a connected graph $\mathcal{H}$, it is easy to verify that $m_i+h_i$ is the same for each vertex $v_i$ in $\mathcal{H}$ if, and only if, $\mathcal{H}$ is a regular graph. Similarly, for a connected bipartite graph $\mathcal{H}$ with partite sets $V_1$ and $V_2$ the quantity $m_i+h_i$ is the same, for all $v_i$ belonging to a given partite set if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

The symmetric division degree index $SDD\left(\mathcal{H}\right)$ of a graph $\mathcal{H}$ is a degree-based graph invariant and is defined as $SDD\left(\mathcal{H}\right)={\displaystyle \sum _{uv\in E\left(\mathcal{H}\right)}}\left({\displaystyle \frac{d_u}{d_v}}+{\displaystyle \frac{d_v}{d_u}}\right)$. The following lower bound for the extended adjacency spectral radius was obtained in [13].

Theorem 7. 

Let $\mathcal{H}$ be a connected graph of order n with symmetric division degree index $SDD\left(\mathcal{H}\right)$. Then,

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{SDD\left(\mathcal{H}\right)}{n}},\end{array}$$

(9)

with equality if, and only if, $m_i+h_i$ is the same for all $v_i\in V\left(\mathcal{H}\right)$. That is, if, and only if, $\mathcal{H}$ is a regular graph.

3. Bounds for the Extended Adjacency Energy

In this section, we derive novel upper and lower bounds for the extended adjacency energy of a graph $\mathcal{H}$, expressed in terms of various structural parameters intrinsic to the graph. We meticulously characterize the extremal graphs that attain these newly established bounds. The focus of our investigation is to explore how these parameters influence the extended adjacency energy and to provide a comprehensive analysis of the conditions under which these bounds are realized. Through detailed examination, we aim to shed light on the interplay between the structural features of the graph and its spectral properties, thereby enhancing our understanding of the extended adjacency energy in different graph configurations.

Let ${\mathcal{M}}_n\left(\mathbb{C}\right)$ be the set of all n-square matrices with complex entries. The Ky Fan k-norm of a matrix $P\in {\mathcal{M}}_n\left(\mathbb{C}\right)$ is defined as ${\parallel P\parallel }_k={\displaystyle {\sum }_{i=1}^k}{\sigma }_i\left(P\right)$, where ${\sigma }_1\left(P\right)\ge {\sigma }_2\left(P\right)\ge \cdots \ge {\sigma }_n\left(P\right)$ are the singular values of P. When $k=1$, the Ky Fan k-norm gives the spectral norm, which is the largest singular value of P. Similarly, when $k=n$, it equals the trace norm, the sum of all singular values of P. For a Hermitian matrix P, the singular values ${\sigma }_i\left(P\right)$ are precisely the absolute values of its eigenvalues $|{\lambda }_i\left(P\right)|$. Hence, in this context, the extended adjacency spectral radius refers to the spectral norm, and the extended adjacency energy corresponds to the trace norm of the matrix $A_{ex}\left(\mathcal{H}\right)$. An intriguing and challenging problem in matrix theory involves determining which matrices within a specified class achieve the maximum and minimum values of the spectral norm and trace norm. The analysis of these norms for matrices associated with graphs and digraphs is extensively documented in the literature, reflecting their importance in various theoretical and applied domains.

The following result provides the extended adjacency ${\mathcal{E}}_{ex}\left(\mathcal{H}\right)$ of a connected graph in terms of the sum of the k largest $A_{ex}$-eigenvalues, which corresponds to the Ky Fan k-norm of the matrix $A_{ex}\left(\mathcal{H}\right)$.

Theorem 8. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$. Then,

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)=2\sum _{{\eta }_i>0}{\eta }_i\left(\mathcal{H}\right)=-2\sum _{{\eta }_i<0}{\eta }_i\left(\mathcal{H}\right)=2\underset{1\le k\le n}{max}\sum _{i=1}^k{\eta }_i\left(\mathcal{H}\right).\end{array}$$

Proof. 

Let $p\left(\mathcal{H}\right)$ be the number of positive eigenvalues and $n\left(\mathcal{H}\right)$ be the number of negative eigenvalues of $A_{ex}\left(\mathcal{H}\right)$. Since ${\displaystyle \sum _{i=1}^n}{\eta }_i\left(\mathcal{H}\right)=0$, it follows that ${\displaystyle \sum _{{\eta }_i>0}}{\eta }_i\left(\mathcal{H}\right)=-{\displaystyle \sum _{{\eta }_i<0}}{\eta }_i\left(\mathcal{H}\right)$. Thus, by the definition of the extended adjacency energy ${\mathcal{E}}_{ex}\left(\mathcal{H}\right)$, we obtain

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)=2{\displaystyle \sum _{{\eta }_i>0}}{\eta }_i\left(\mathcal{H}\right)=-2{\displaystyle \sum _{{\eta }_i<0}}{\eta }_i\left(\mathcal{H}\right).\end{array}$$

(10)

This proves the first and the second equalities. From (10), we have ${\mathcal{E}}_{ex}\left(\mathcal{H}\right)=2{\displaystyle \sum _{{\eta }_i>0}}{\eta }_i\left(\mathcal{H}\right)=2{\displaystyle \sum _{i=1}^{p\left(\mathcal{H}\right)}}{\eta }_i\left(\mathcal{H}\right)$. Let k be a positive integer with $1\le k\le n$. To prove the third equality, it suffices to show that ${\displaystyle \sum _{i=1}^{p\left(\mathcal{H}\right)}}{\eta }_i\ge {\displaystyle \sum _{i=1}^k}{\eta }_i$, for all k. If $k\le p\left(\mathcal{H}\right)$, then by using ${\eta }_i>0$, for $i=1,2,\cdots ,p\left(\mathcal{H}\right)$, it follows that ${\displaystyle \sum _{i=1}^{p\left(\mathcal{H}\right)}}{\eta }_i\left(\mathcal{H}\right)\ge {\displaystyle \sum _{i=1}^k}{\eta }_i\left(\mathcal{H}\right)$. On the other hand, if $k>p\left(\mathcal{H}\right)$, then ${\displaystyle \sum _{i=1}^k}{\eta }_i={\displaystyle \sum _{i=1}^{p\left(\mathcal{H}\right)}}{\eta }_i\left(\mathcal{H}\right)+{\displaystyle \sum _{i=p\left(\mathcal{H}\right)+1}^k}{\eta }_i\left(\mathcal{H}\right)<{\displaystyle \sum _{i=1}^{p\left(\mathcal{H}\right)}}{\eta }_i\left(\mathcal{H}\right)$, as ${\eta }_i\le 0$, for $i\ge p\left(\mathcal{H}\right)+1$. Thus, in each case, we have ${\displaystyle \sum _{i=1}^{p\left(\mathcal{H}\right)}}{\eta }_i\left(\mathcal{H}\right)\ge {\displaystyle \sum _{i=1}^k}{\eta }_i\left(\mathcal{H}\right)$. This completes the proof. □

The following observation immediately follows from Theorem 8: the extended adjacency energy of a graph $\mathcal{H}$ is at least twice the extended adjacency spectral radius.

Theorem 9. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$. Then, ${\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge 2{\eta }_1\left(\mathcal{H}\right)$, with equality if, and only if, $\mathcal{H}$ has one positive extended adjacency eigenvalue.

Proof. 

From Theorem 8, we have

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)=2\underset{1\le k\le n}{max}\sum _{i=1}^k{\eta }_i\left(\mathcal{H}\right)\ge 2{\eta }_1\left(\mathcal{H}\right).\end{array}$$

(11)

Equality occurs in (11) if, and only if, $\mathcal{H}$ has one positive $A_{ex}$-eigenvalue. This completes the proof. □

Using Theorem 1 together with inequality (11), we arrive at the following result:

Corollary 4. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$. Then,

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge 2\lambda \left(A\left(\mathcal{H}\right)\right)+2\epsilon ,\end{array}$$

where $\epsilon \ge 0$ is given by $\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\left\{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)\right\}=1+\epsilon$. For $\epsilon =0$, equality occurs if, and only if, $\mathcal{H}$ is a regular complete multipartite graph. For $\epsilon >0$, equality occurs if, and only if, $\mathcal{H}$ is a complete bipartite graph with partite sets of distinct degrees.

Proof. 

The proof follows directly from the inequality (11) by using Theorem 1. Equality occurs if, and only if, equality holds in Theorem 1 and equality holds in (11). Specifically, for $\epsilon =0$, equality in Theorem 1 holds if, and only if, $\mathcal{H}$ is a regular graph, and for $\epsilon >0$, equality holds if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph.

Moreover, equality in (11) occurs if, and only if, $\mathcal{H}$ has exactly one positive extended adjacency eigenvalue. Combining these observations, we conclude that for $\epsilon =0$, equality occurs if, and only if, $\mathcal{H}$ is a regular graph with one positive extended adjacency eigenvalue, and for $\epsilon >0$, equality occurs if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph with one positive extended adjacency eigenvalue.

Since for regular graphs $A_{ex}\left(\mathcal{H}\right)=A\left(\mathcal{H}\right)$ and for bipartite $(r_1,r_2)$-semi-regular graphs $A_{ex}\left(\mathcal{H}\right)={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{r_1}{r_2}}+{\displaystyle \frac{r_2}{r_1}}\right)A\left(\mathcal{H}\right)$, it follows that a regular graph or a bipartite semi-regular graph $\mathcal{H}$ has one positive extended adjacency eigenvalue if, and only if, $\mathcal{H}$ has one positive eigenvalue.

Using the well-known fact (see [1]) that a graph $\mathcal{H}$ has one positive eigenvalue if, and only if, $\mathcal{H}$ is a complete multipartite graph, we conclude that for $\epsilon =0$, equality occurs if, and only if, $\mathcal{H}$ is a regular complete multipartite graph, and for $\epsilon >0$, equality occurs if, and only if, $\mathcal{H}$ is a complete bipartite graph. This completes the proof. □

The following observation is immediate from Corollary 4:

Corollary 5. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$.

(i).

If m is the number of edges, then

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge {\displaystyle \frac{4m}{n}}+2\epsilon ,\end{array}$$

with quality if, and only if, $\epsilon =0$ and $\mathcal{H}$ is a regular complete multipartite graph.

(ii).

If $M_1$ is the first Zagreb index of $\mathcal{H}$, then

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge 2\sqrt{{\displaystyle \frac{M_1}{n}}}+2\epsilon ,\end{array}$$

with equality for $\epsilon =0$, if, and only if, $\mathcal{H}$ is a regular complete multipartite graph and equality for $\epsilon >0$, if, and only if, $\mathcal{H}$ is a complete bipartite graph with partite sets consisting of distinct degrees.

(iii).

If $SDD\left(\mathcal{H}\right)$ is the symmetric division degree index, then

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge {\displaystyle \frac{2SDD\left(\mathcal{H}\right)}{n}},\end{array}$$

with equality if, and only if, $\mathcal{H}$ is a regular complete multipartite graph.

(iv).

If $d_i$ is the degree of the vertex $v_i\in V\left(\mathcal{H}\right)$, then

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge 2(1+\epsilon )\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\sqrt{d_i{d}_j},\end{array}$$

with equality for $\epsilon =0$, if, and only if, $\mathcal{H}$ is a regular complete multipartite graph and equality for $\epsilon >0$, if, and only if, $\mathcal{H}$ is a complete bipartite graph with partite sets consisting of distinct degrees.

Proof. 

The proof of (i) and (ii) follows directly from Corollary 4 by using the fact that $\lambda \left(A\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{2m}{n}}$, with equality if, and only if, $\mathcal{H}$ is a regular graph and $\lambda \left(A\left(\mathcal{H}\right)\right)\ge \sqrt{{\displaystyle \frac{M_1}{n}}}$, with equality if, and only if, $\mathcal{H}$ is a regular graph or $\mathcal{H}$ is a bipartite semi-regular graph. The proof of (iii) follows by using Theorem 7 together with Theorem 9. Equality occurs if, and only if, equality occurs in Theorem 9 and equality occurs in Theorem 7. Equality occurs in Theorem 9 if, and only if, $\mathcal{H}$ has one positive extended adjacency eigenvalue and equality occurs in Theorem 7 if, and only if, $\mathcal{H}$ is a regular graph. Combining these statements, we conclude that equality occurs in (iii) if, and only if, $\mathcal{H}$ is a regular graph with one positive extended adjacency eigenvalue. Since a regular graph has one positive extended adjacency eigenvalue if, and only if, it has one positive adjacency eigenvalue and using a well-known fact [1] that a graph has one positive adjacency eigenvalue if, and only if, it is a complete multipartite graph, it follows that equality occurs in (iii) if, and only if, $\mathcal{H}$ is a complete multipartite graph. Similarly, using Theorem 6 together with Theorem 9, we arrive at (iv). The equality case for (iv) follows by using the fact that equality occurs in Theorem 6 if, and only if, $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph together with the fact that a graph $\mathcal{H}$ has one positive eigenvalue if, and only if, $\mathcal{H}$ is a complete multipartite graph. This completes the proof. □

The lower bound given by (iii) was obtained in [13] but the equality case was not discussed completely. Here, we have completely characterized the graphs which attain this lower bound.

It is clear that our lower bound given by (i) of Corollary 5 for the extended adjacency energy is always better than the lower bound

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\ge {\displaystyle \frac{4m}{n}},\end{array}$$

obtained in [13]. Moreover, the equality case was not discussed in [13]. Here, we have completely characterized the graphs which attain this lower bound.

One of the well-studied problems in spectral graph theory is determining the extremal values of certain spectral invariants and characterizing the graphs that achieve these extremal values among a class of graphs of order n. This problem has received extensive attention, particularly for the spectral radius and energy of various graph matrices, with numerous papers dedicated to this theme in the literature.

The following observation is evident from Theorem 9:

Corollary 6. 

Among all connected graphs of order n, the graphs with exactly one positive extended adjacency eigenvalue have the minimum extended adjacency energy, which equals $2\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)$.

Therefore, it will be an interesting problem for future research to characterize the graphs with just one positive extended adjacency eigenvalue.

The following lemma gives a version of the arithmetic-geometric mean inequality; see [19].

Lemma 2. 

If $w_1,w_2,\cdots ,w_n$ are non-negative numbers, then

$$\begin{array}{cc}\hfill n\left[{\displaystyle \frac{1}{n}}\sum _{j=1}^n{w}_j-{\left(\prod _{j=1}^n{w}_j\right)}^{{\displaystyle \frac{1}{n}}}\right]& \le n\sum _{j=1}^n{w}_j-{\left(\sum _{j=1}^n\sqrt{w_j}\right)}^2\hfill \\ & \le n(n-1)\left[{\displaystyle \frac{1}{n}}\sum _{j=1}^n{w}_j-{\left(\prod _{j=1}^n{w}_j\right)}^{{\displaystyle \frac{1}{n}}}\right].\hfill \end{array}$$

Moreover, equality occurs if, and only if, $w_1=w_2=\cdots =w_n$.

The Frobenius norm of a $p\times q$ matrix $C=\left(m_{ij}\right)$ is denoted by ${\parallel C\parallel }_F$ and it is defined as

$${\parallel C\parallel }_F=\sqrt{\sum _{i=1}^p\sum _{j=1}^q{\left|m_{ij}\right|}^2}=\sqrt{tr\left(C^{T}C\right)}=\sqrt{\sum _{i=1}^{min\{p,q\}}{\sigma }_i^2\left(C\right)},$$

where ${\sigma }_i\left(C\right)$ is the i-th singular value of C. If $C^T=C$, a symmetric matrix, then ${\sigma }_i\left(C\right)=\left|{\lambda }_i\left(C\right)\right|$, where ${\lambda }_i\left(C\right)$ is the i-th eigenvalue of C and so ${\parallel C\parallel }_F^2={\displaystyle \sum _{i=1}^n}{\lambda }_i{\left(C\right)}^2=tr\left(C^2\right)$, where $tr\left(C\right)$ denotes the trace of C.

The following result provides upper and lower bounds for the extended adjacency energy in terms of the order n, the parameter $\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2$, and the determinant of the matrix $A_{ex}\left(\mathcal{H}\right)$:

Theorem 10. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$. Let $m_i$ be the average 2-degree and $h_i$ be the harmonic average 2-degree of the vertex $v_i$. Then,

$$\begin{array}{cc}\hfill {\beta }_2+\sqrt{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_1^2+(n-2){{\beta }_1}^{{\displaystyle \frac{-2}{n-1}}}\gamma }\le & {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\hfill \\ \hfill \le & {\beta }_1+\sqrt{(n-2)(\parallel A_{ex}\left(\mathcal{H}\right){\parallel }_F^2-{\beta }_2^2)+\gamma {{\beta }_2}^{{\displaystyle \frac{-2}{n-1}}}},\hfill \end{array}$$

(12)

where $\gamma =(n-1)|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}},{\beta }_1={\displaystyle \frac{1}{2}}\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{max}\sqrt{(m_i+h_i)(m_j+h_j)}$ and ${\beta }_2$$={\displaystyle \frac{1}{2}}\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{min}\sqrt{(m_i+h_i)(m_j+h_j)}$. Equality occurs if, and only if, $\mathcal{H}\cong K_n$ or $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph with three distinct eigenvalues, ${\beta }_1$ and the other two eigenvalues with absolute value $\sqrt{{\displaystyle \frac{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_1^2}{n-1}}}$.

Proof. 

By replacing n by $n-1$ and setting $w_j={\left|{\eta }_j\right|}^2,$ for $j=2,\cdots ,n$ in Lemma 2, we obtain

$$\begin{array}{c}\hfill \alpha \le (n-1)\sum _{j=2}^n{\left|{\eta }_j\right|}^2-{\left(\sum _{j=2}^n\left|{\eta }_j\right|\right)}^2\le (n-2)\alpha ,\end{array}$$

that is,

$$\begin{array}{c}\hfill \alpha \le (n-1)\sum _{j=2}^n{\left|{\eta }_j\right|}^2-{\left({\mathcal{E}}_{ex}\left(\mathcal{H}\right)-{\eta }_1\right)}^2\le (n-2)\alpha ,\end{array}$$

(13)

where

$$\begin{array}{cc}\hfill \alpha & =(n-1)\left[{\displaystyle \frac{1}{n-1}}\sum _{j=2}^n{\left|{\eta }_j\right|}^2-{\left(\prod _{j=2}^n{\left|{\eta }_j\right|}^2\right)}^{{\displaystyle \frac{1}{n-1}}}\right]\hfill \\ & =\sum _{j=2}^n{\left|{\eta }_j\right|}^2-(n-1){\left(\prod _{j=2}^n\left|{\eta }_j\right|\right)}^{{\displaystyle \frac{2}{n-1}}}\hfill \\ & =\sum _{j=2}^n{\left|{\eta }_j\right|}^2-(n-1){\eta }_1^{{\displaystyle \frac{-2}{n-1}}}|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}}.\hfill \end{array}$$

By using ${\displaystyle \sum _{j=2}^n}|{\eta }_j{|}^2={\parallel A_{ex}\left(\mathcal{H}\right)\parallel }_F^2-{\eta }_1^2$ and the value of $\alpha$, it follows from the left inequality of (13) that

$$\begin{array}{c}\hfill {\left({\mathcal{E}}_{ex}\left(\mathcal{H}\right)-{\eta }_1\right)}^2\le (n-2)\sum _{j=2}^n{\left|{\eta }_j\right|}^2+(n-1){\eta }_1^{{\displaystyle \frac{-2}{n-1}}}|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}},\end{array}$$

that is,

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\le {\eta }_1+\sqrt{(n-2)(\parallel A_{ex}\left(\mathcal{H}\right){\parallel }_F^2-{\eta }_1^2)+(n-1){\eta }_1^{{\displaystyle \frac{-2}{n-1}}}|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}}}.\end{array}$$

(14)

Again by using ${\displaystyle \sum _{j=2}^n}|{\eta }_j{|}^2={\parallel A_{ex}\left(\mathcal{H}\right)\parallel }_F^2-{\eta }_1^2$ and the value of $\alpha$, it follows from the right inequality of (13) that

$$\begin{array}{c}\hfill {\left({\mathcal{E}}_{ex}\left(\mathcal{H}\right)-{\eta }_1\right)}^2\ge \sum _{j=2}^n{\left|{\eta }_j\right|}^2+(n-1)(n-2){\eta }_1^{{\displaystyle \frac{-2}{n-1}}}|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}},\end{array}$$

that is,

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathcal{H}\right)\ge {\eta }_1+\sqrt{{\parallel A}_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\eta }_1^2+(n-1)(n-2){\eta }_1^{{\displaystyle \frac{-2}{n-1}}}|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}}}.\end{array}$$

(15)

Now, using ${\eta }_1=\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\beta }_1$ and ${\eta }_1=\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge {\beta }_2$ in (14) and (15), we obtain the right and the left inequalities in (12).

Equality occurs in (12) if, and only if, equality occurs in Lemma 2 and equality occurs in Theorem 3 and in Theorem 5. Since equality occurs in Lemma 2 if, and only if, $|{\eta }_2{|}^2=|{\eta }_3{|}^2=\cdots ={\left|{\eta }_n\right|}^2$, that is, if, and only if, $|{\eta }_2|=|{\eta }_3|=\cdots =|{\eta }_n|$, as ${\eta }_i$ are real numbers. Also, equality occurs in Theorems 3 and 5 if, and only if, $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph. We conclude that equality occurs in (12) if, and only if, $\mathcal{H}$ is a regular graph with $|{\eta }_2|=|{\eta }_3|=\cdots =|{\eta }_n|$ or equality occurs in (12) if, and only if, $\mathcal{H}$ is a bipartite semi-regular graph with $|{\eta }_2|=|{\eta }_3|=\cdots =|{\eta }_n|$. The following cases arise.

Case 1. If ${\eta }_2={\eta }_3=\cdots ={\eta }_n$, then $\mathcal{H}$ is a connected graph with two distinct extended adjacency eigenvalues and so by Theorem 11 of [13], it follows that $\mathcal{H}\cong K_n$.

Case 2. If ${\eta }_i\ne {\eta }_j$, for at least one $i,j$, then there exists a positive integer k, such that ${\eta }_2=\cdots ={\eta }_k=\theta$ and ${\eta }_{k+1}=\cdots ={\eta }_n=-\theta$. Since ${\displaystyle \sum _{j=2}^n}{\eta }_j^2={\parallel A_{ex}\left(\mathcal{H}\right)\parallel }_F^2-{\eta }_1^2$, gives that $(n-1){\theta }^2={\displaystyle \sum _{j=2}^k}{\eta }_j^2+{\displaystyle \sum _{j=k+1}^n}{\eta }_j^2={\parallel A_{ex}\left(\mathcal{H}\right)\parallel }_F^2-{\eta }_1^2$, it follows that equality occurs in this case if $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph with three distinct extended adjacency eigenvalues, namely, the eigenvalue ${\beta }_1$ and the other two eigenvalues with absolute value $\sqrt{{\displaystyle \frac{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_1^2}{n-1}}}$. Now, by using the fact that if $\mathcal{H}$ is a regular graph or a bipartite semi-regular graph, then $\mathcal{H}$ has three distinct extended adjacency eigenvalues if, and only if, $\mathcal{H}$ has three distinct adjacency eigenvalues. Thus, the result follows.

Conversely, it is straightforward to verify that equality holds in (12) for each of the mentioned cases. This completes the proof. □

By using $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)\lambda \left(A\left(\mathcal{H}\right)\right)$ and $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge (1+\epsilon )\lambda \left(A\left(\mathcal{H}\right)\right)$ in the proof of Theorem 10, we arrive at the following result:

Theorem 11. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$ with maximum degree Δ and minimum degree δ. Then,

$$\begin{array}{cc}\hfill {\beta }_4+\sqrt{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_3^2+(n-2){{\beta }_3}^{{\displaystyle \frac{-2}{n-1}}}\gamma }\le & {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\hfill \\ \hfill \le & {\beta }_3+\sqrt{(n-2)(\parallel A_{ex}\left(\mathcal{H}\right){\parallel }_F^2-{\beta }_4^2)+\gamma {{\beta }_4}^{{\displaystyle \frac{-2}{n-1}}}},\hfill \end{array}$$

where $\gamma =(n-1)|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}}$, ${\beta }_3={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)\lambda \left(A\left(\mathcal{H}\right)\right)$ and ${\beta }_4=(1+\epsilon )\lambda \left(A\left(\mathcal{H}\right)\right)$. Equality occurs if, and only if, $\mathcal{H}\cong K_n$ or $\mathcal{H}$ is a regular graph (for $\epsilon =0$) or a bipartite semi-regular graph (for $\epsilon >0)$ with three distinct eigenvalues, ${\beta }_3$ and the other two eigenvalues with absolute value $\sqrt{{\displaystyle \frac{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_3^2}{n-1}}}$.

By using $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)\lambda \left(A\left(\mathcal{H}\right)\right)$ and $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{SDD\left(\mathcal{H}\right)}{n}}$ in the proof of Theorem 10, we arrive at the following result:

Theorem 12. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$. Let Δ be the maximum degree and δ be the minimum degree of $\mathcal{H}$. Then,

$$\begin{array}{cc}\hfill {\beta }_5+\sqrt{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_3^2+(n-2){{\beta }_3}^{{\displaystyle \frac{-2}{n-1}}}\gamma }\le & {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\hfill \\ \hfill \le & {\beta }_3+\sqrt{(n-2)(\parallel A_{ex}\left(\mathcal{H}\right){\parallel }_F^2-{\beta }_5^2)+\gamma {{\beta }_5}^{{\displaystyle \frac{-2}{n-1}}}},\hfill \end{array}$$

where $\gamma =(n-1)|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}},{\beta }_3={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\Delta }{\delta }}+{\displaystyle \frac{\delta }{\Delta }}\right)\lambda \left(A\left(\mathcal{H}\right)\right)$ and ${\beta }_5={\displaystyle \frac{SDD\left(\mathcal{H}\right)}{n}}$. Equality occurs if, and only if, $\mathcal{H}\cong K_n$ or $\mathcal{H}$ is a regular graph with three distinct eigenvalues, ${\beta }_3$ and the other two eigenvalues with absolute value $\sqrt{{\displaystyle \frac{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_3^2}{n-1}}}$.

By using $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\le {\displaystyle \frac{1}{2}}\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{max}\sqrt{(m_i+h_i)(m_j+h_j)}$ and $\lambda \left(A_{ex}\left(\mathcal{H}\right)\right)\ge {\displaystyle \frac{SDD\left(\mathcal{H}\right)}{n}}$ in the proof of Theorem 10, we arrive at the following result:

Theorem 13. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$. Let $m_i$ be the average 2-degree and $h_i$ be the harmonic average 2-degree of the vertex $v_i$. Then,

$$\begin{array}{cc}\hfill {\beta }_5+\sqrt{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_1^2+(n-2){{\beta }_1}^{{\displaystyle \frac{-2}{n-1}}}\gamma }\le & {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\hfill \\ \hfill \le & {\beta }_1+\sqrt{(n-2)(\parallel A_{ex}\left(\mathcal{H}\right){\parallel }_F^2-{\beta }_5^2)+\gamma {{\beta }_5}^{{\displaystyle \frac{-2}{n-1}}}},\hfill \end{array}$$

where $\gamma =(n-1)|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n-1}}},{\beta }_1={\displaystyle \frac{1}{2}}\underset{v_i{v}_j\in E\left(\mathcal{H}\right)}{max}\sqrt{(m_i+h_i)(m_j+h_j)}$ and ${\beta }_5={\displaystyle \frac{SDD\left(\mathcal{H}\right)}{n}}$. Equality occurs if, and only if, $\mathcal{H}\cong K_n$ or $\mathcal{H}$ is a regular graph with three distinct eigenvalues, ${\beta }_1$ and the other two eigenvalues with absolute value $\sqrt{{\displaystyle \frac{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2-{\beta }_1^2}{n-1}}}$.

By setting $w_j={\left|{\eta }_j\right|}^2,$ for $j=1,2,\cdots ,n$ in Lemma 2 and proceeding similar to the proof of Theorem 10, we arrive at the following result for the extended adjacency energy:

Theorem 14. 

Let $\mathcal{H}$ be a connected graph of order $n\ge 3$ and let $det\left(A_{ex}\left(\mathcal{H}\right)\right)$ be the determinant of $A_{ex}\left(\mathcal{H}\right)$. Then,

$$\begin{array}{c}\hfill \sqrt{\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2+(n-1){\gamma }_1}\le {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\le \sqrt{(n-1)\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2+{\gamma }_1},\end{array}$$

(16)

where $\gamma =n|det\left(A_{ex}\left(\mathcal{H}\right)\right){|}^{{\displaystyle \frac{2}{n}}}$. Equality occurs if, and only if, $\mathcal{H}\cong K_2$.

Proof. 

The result follows from Lemma 2 by employing the same technique as in Theorem 10. Equality occurs in (16) if, and only if, equality occurs in Lemma 2 in the case where all $a_i^{\prime }s$ are equal. This gives that equality occurs in (16) if, and only if, $|{\eta }_1{|}^2=|{\eta }_2{|}^2=\cdots ={\left|{\eta }_n\right|}^2$, that is, if, and only if, $|{\eta }_1|=|{\eta }_2|=\cdots =|{\eta }_n|$. Now, by using Lemma 7 from [9], the equality case follows. □

The following upper bound for the extended adjacency energy was reported in [11]:

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left(\mathcal{H}\right)\le \sqrt{n\parallel A_{ex}{\left(\mathcal{H}\right)\parallel }_F^2}.\end{array}$$

(17)

By applying the arithmetic-geometric mean inequality, it can be observed that the upper bound provided in Theorem 14 is always superior to the upper bound given by (17).

Example 1. 

Consider the graph ${\mathcal{H}}_1$ with vertices $v_1,v_2,v_3,v_4,v_5$ and edges $v_1{v}_2,v_2{v}_3,v_3{v}_4,v_3{v}_5$ as shown in the figure. For this graph, we have $n=5;m=4;d_1=1,d_2=2,d_3=3,d_4=1,$$d_5=1;M_1=16;SO\left({\mathcal{H}}_1\right)=12.1661;SDD\left({\mathcal{H}}_1\right)=11.3333$, also ${min}_{v_i{v}_j\in E\left({\mathcal{H}}_1\right)}\left({\displaystyle \frac{d_i}{d_j}}+{\displaystyle \frac{d_j}{d_i}}\right)={\displaystyle \frac{13}{12}}=1+{\displaystyle \frac{1}{12}}$ giving that $\epsilon ={\displaystyle \frac{1}{12}}$. Further, $m_1+h_1={\displaystyle \frac{5}{2}},m_2+h_2={\displaystyle \frac{14}{3}},m_3+h_3={\displaystyle \frac{53}{6}},m_4+h_4={\displaystyle \frac{10}{3}}$ and $m_5+h_5={\displaystyle \frac{10}{3}}$. The extended adjacency matrix of ${\mathcal{H}}_1$ is $A_{ex}\left({\mathcal{H}}_1\right)=\left(\begin{array}{ccccc}0& {\displaystyle \frac{5}{4}}& 0& 0& 0\\ {\displaystyle \frac{5}{4}}& 0& {\displaystyle \frac{13}{12}}& 0& 0\\ 0& {\displaystyle \frac{13}{12}}& 0& {\displaystyle \frac{5}{3}}& {\displaystyle \frac{5}{3}}\\ 0& 0& {\displaystyle \frac{5}{3}}& 0& 0\\ 0& 0& {\displaystyle \frac{5}{3}}& 0& 0\end{array}\right)$. By direct calculation, it can be seen that the eigenvalues of $A_{ex}\left({\mathcal{H}}_1\right)$ are $0,\pm 1.1086,\pm 2.6575$ giving that ${\lambda }_1\left(A_{ex}\left({\mathcal{H}}_1\right)\right)=2.6575,det(A_{ex}\left({\mathcal{H}}_1\right)=0,\parallel A_{ex}{\left({\mathcal{H}}_1\right|}_F^2=16.6895$ and ${\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)=7.5322$. Again by direct calculation, it can be seen that the adjacency spectral radius of ${\mathcal{H}}_1$ is ${\lambda }_1\left({\mathcal{H}}_1\right)=1.8477$. Now, let us see the application of the results obtained in Section 2 and Section 3 for the extended adjacency spectral radius and the energy of the graph ${\mathcal{H}}_1$. The estimates given by 1 give

$$\begin{array}{c}\hfill 1.8477\approx \sqrt{\sqrt{2}+2}\le \lambda \left(A_{ex}\left({\mathcal{H}}_1\right)\right)\le {\displaystyle \frac{5}{3}}\sqrt{\sqrt{2}+2}\approx 3.0795.\end{array}$$

The lower bound given by Corollary 1 gives

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left({\mathcal{H}}_1\right)\right)\ge {\displaystyle \frac{13}{12}}\sqrt{\sqrt{2}+2}\approx 2.0017.\end{array}$$

The lower bound given by Corollary 3 gives

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left({\mathcal{H}}_1\right)\right)\ge \sqrt{{\displaystyle \frac{1}{9}}+{\displaystyle \frac{1}{4}}}{\displaystyle \frac{12.1661}{5}}\approx 1.4621.\end{array}$$

The upper bound given by Theorem 3 gives

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left({\mathcal{H}}_1\right)\right)\le {\displaystyle \frac{1}{2}}max\{\sqrt{{\displaystyle \frac{35}{3}}},\sqrt{{\displaystyle \frac{371}{9}}},\sqrt{{\displaystyle \frac{265}{9}}}\}={\displaystyle \frac{\sqrt{371}}{6}}\approx 3.2102.\end{array}$$

The lower bound given by Theorem 5 gives

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left({\mathcal{H}}_1\right)\right)\ge {\displaystyle \frac{1}{2}}min\{\sqrt{{\displaystyle \frac{35}{3}}},\sqrt{{\displaystyle \frac{371}{9}}},\sqrt{{\displaystyle \frac{265}{9}}}\}={\displaystyle \frac{\sqrt{35}}{2\sqrt{3}}}\approx 1.7078.\end{array}$$

The lower bound given by Theorem 7 gives

$$\begin{array}{c}\hfill \lambda \left(A_{ex}\left({\mathcal{H}}_1\right)\right)\ge {\displaystyle \frac{34}{15}}\approx 2.2666.\end{array}$$

The lower bound given by Corollary 4 gives

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\ge 2\sqrt{\sqrt{2}+2}+{\displaystyle \frac{1}{6}}\approx 3.8621.\end{array}$$

The lower bounds given by parts (i), (ii), (iii), and (iv) of the Corollary 5 give

$$\begin{array}{c}\hfill {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\ge {\displaystyle \frac{16}{5}}+{\displaystyle \frac{1}{6}}\approx 3.3666,\\ \hfill {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\ge 2\sqrt{{\displaystyle \frac{16}{5}}}+{\displaystyle \frac{1}{6}}\approx 3.7443,\\ \hfill {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\ge {\displaystyle \frac{68}{15}}\approx 4.5333,\\ \hfill {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\ge {\displaystyle \frac{26}{12}}\sqrt{2}\approx 3.0641.\end{array}$$

The estimates given by Theorem 10 give

$$\begin{array}{c}\hfill 4.2344\approx {\displaystyle \frac{\sqrt{35}}{2\sqrt{3}}}+\sqrt{16.6895-{\displaystyle \frac{371}{36}}}\le {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right){\displaystyle \frac{\sqrt{371}}{6}}+\sqrt{3(16.6895-{\displaystyle \frac{35}{12}})}\approx 9.6381.\end{array}$$

The bounds given by Theorem 11 give

$$\begin{array}{cc}\hfill 4.6861\approx 2.0017+\sqrt{16.6895-{\left(3.0795\right)}^2}\le & {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\le \hfill \\ & 3.0795+\sqrt{3(16.6895-{\left(2.0017\right)}^2)}\approx 9.2478.\hfill \end{array}$$

The bounds given by Theorem 12 give

$$\begin{array}{cc}\hfill 4.9511\approx {\displaystyle \frac{11.3333}{5}}+\sqrt{16.6895-{\left(3.0795\right)}^2}\le & {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\le \hfill \\ & 3.0795+\sqrt{3(16.6895-{\left({\displaystyle \frac{11.3333}{5}}\right)}^2)}\approx 8.9663.\hfill \end{array}$$

The estimates obtained in Theorem 13 give

$$\begin{array}{cc}\hfill 4.7933\approx {\displaystyle \frac{11.3333}{5}}+\sqrt{16.6895-{\left(3.2102\right)}^2}\le & {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\le \hfill \\ & 3.2102+\sqrt{3(16.6895-{\left({\displaystyle \frac{11.3333}{5}}\right)}^2)}\approx 9.0970.\hfill \end{array}$$

Finally, the bounds obtained in Theorem 14 give

$$\begin{array}{c}\hfill 4.0852\approx \sqrt{16.6895}\le {\mathcal{E}}_{ex}\left({\mathcal{H}}_1\right)\le 2\sqrt{16.6895}\approx 8.1705.\end{array}$$

From Example 1, it is clear that the bounds obtained in Section 2 for the extended adjacency spectral radius and the bounds obtained in Section 3 for the extended adjacency energy are, in general, incomparable.

4. Conclusions

This work is an effort to derive some new upper and lower bounds for the extended adjacency spectral radius and the extended adjacency energy of a graph. Our motivation was to introduce some new graph parameters and study their connection with these spectral graph invariants. Many potential directions in this field can be explored for future research.