Eccentricity Based Topological Indices of an Oxide Network

Abstract

Graph theory has much great advances in the field of mathematical chemistry. Chemical graph theory has become very popular among researchers because of its wide applications in mathematical chemistry. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. A great variety of such indices are studied and used in theoretical chemistry, pharmaceutical researchers, in drugs and in different other fields. In this article, we study the chemical graph of an oxide network and compute the total eccentricity, average eccentricity, eccentricity based Zagreb indices, atom-bond connectivity ($ABC$) index and geometric arithmetic index of an oxide network. Furthermore, we give analytically closed formulas of these indices which are helpful in studying the underlying topologies.

1. Introduction

Graph theory is a branch of mathematics that has a lot of applications in computer science, electrical systems (network), interconnected systems (network), biological networks, and in chemistry. Chemical graph theory is the rapidly developing zone among chemists and mathematicians. Chemical graph theory helps us to predict the certain physico-chemical properties of chemical compounds by just considering their pictorial representations [1,2].

Cheminformatics is a comparatively new subject, which is a combination of chemistry, mathematics and information science. There is a considerable usage of graph theory in theoretical and computational chemistry. Chemical graph theory is the topology branch of mathematical chemistry which implements graph theory to mathematically model chemical occurrences. There has been a lot of research in this area in the last few decades. A few references are given that demonstrate the significance of graph theory in Mathematical Chemistry [3,4].

Let $G=(V,E)$ be a graph, where V is a non-empty set of vertices and E is a set of edges. Chemical graph theory applies graph theory to the mathematical modeling of molecular phenomena, which is helpful for the study of molecular structures. The manipulation and examination of chemical structural information is made conceivable by using molecular descriptors. A great variety of topological indices are studied and used in theoretical chemistry by pharmaceutical researchers. In chemical graph theory, there are many topological indices for a connected graph, which are helpful in the study of chemical molecules. This theory has had a great effect in the development of chemical science.

If $p,q\in V\left(G\right)$, then the distance $d(p,q)$ between p and q is defined as the length of any shortest path in G connecting p and q. Eccentricity is the distance of vertex u from the farthest vertex in G. In mathematical form,

$$\epsilon \left(u\right)=max\left\{d\right(u,v\left)\right|\forall u\in V\left(G\right)\}.$$

(1)

The total eccentricity index is introduced by Farooq et al. [5], which is defined as,

$$\zeta \left(G\right)=\sum _{v\in V\left(G\right)}\epsilon \left(v\right).$$

(2)

where $\epsilon \left(v\right)$ represents eccentricity of vertex v.

The average eccentricity $avec\left(G\right)$ of a graph G is the mean value of eccentricities of all vertices of a graph, that is,

$$avec\left(G\right)=\frac{1}{n}\sum _{v\in V\left(G\right)}\epsilon \left(v\right).$$

(3)

The average eccentricity and standard deviation for all Sierpiński graphs $S_p^n$ is established by [6]. The extremal properties of the average eccentricity, conjectures and Auto graphicx, about the average eccentricity are obtained by [7]. The bounds on the mean eccentricity of a graph, and also the change in mean eccentricity when a graph is replaced by a subgraph, is established by [8]. For trees with fixed diameter, fixed matching number and fixed number of pendent vertices, the lower and upper bounds of average eccentricity are found by [9].

The “eccentricity based geometric-arithmetic ($GA$)” index of a graph G is defined as [10],

$$G{A}_4\left(G\right)=\sum _{uvϵE\left(G\right)}\frac{2\sqrt{\epsilon \left(u\right)·\epsilon \left(v\right)}}{\epsilon \left(u\right)+\epsilon \left(v\right)}.$$

(4)

Further results regarding the average eccentricity index and eccentricity-based geometric-arithmetic index can be found in [11]. A new version of the $ABC$ index is introduced by Farahani [12] which is defined as,

$$AB{C}_5\left(G\right)=\sum _{uv\in E\left(G\right)}\sqrt{\frac{\epsilon \left(v\right)+\epsilon \left(u\right)-2}{\epsilon \left(v\right)·\epsilon \left(u\right)}}.$$

(5)

Imran et al. computed the eccentricity based $ABC$ index and eccentricity based geometric-arithmetic index for copper oxide in [13]. Gao et al. calculated the result about the eccentric $ABC$ index of linear polycene parallelogram benzenoid in [14].

In 2010, D. Vukičević et al. and in 2012, Ghorbani et al. proposed some new modified versions of Zagreb indices of a molecular graph G [15,16]. The first Zagreb eccentricity index is defined as:

$$M_1^{*}\left(G\right)=\sum _{uv\in E\left(G\right)}[\epsilon \left(u\right)+\epsilon \left(v\right)].$$

(6)

The second Zagreb eccentricity index is defined as:

$$M_1^{**}\left(G\right)=\sum _{v\in V\left(G\right)}{\left[\epsilon \left(v\right)\right]}^2.$$

(7)

The third Zagreb eccentricity index is defined as:

$$M_2^{*}\left(G\right)=\sum _{uv\in E\left(G\right)}\epsilon \left(u\right)\epsilon \left(v\right).$$

(8)

So, in this article, we extend the study of chemical graph theory to compute the total eccentricity, average eccentricity, eccentricity-based Zagreb indices, $ABC$ index and geometric arithmetic index of oxide network. Furthermore, we give the exact result of these indices which are helpful in studying the underlying topological properties of oxide networks.

2. Applications of Topological Indices and Motivation

The $ABC$ index provides a very good correlation for the stability of linear alkanes as well as the branched alkanes and for computing the strain energy of cyclo alkanes [17,18,19,20]. To correlate with certain physico-chemical properties, the $GA$ index has much better predictive power than the predictive power of the Randic connectivity index [21,22,23]. The first and second Zagreb index were found to occur for computation of the total $\pi$-electron energy of the molecules within specific approximate expressions [24].

Since degree based topological indices are useful to analyzed the chemical properties of different molecular structures. So motivated by this idea, we focus on eccentricity based topological indices. As eccentricity based topological indices are used as an important tool to the prediction of physico-chemical, pharmacological and toxicological properties of a compound directly from its molecular structure. This analysis is known as the study of the quantitative structure–activity relationship (QSAR) [25].

3. Methods

To compute our results, we use the method of combinatorial computing, vertex partition method, edge partition method, graph theoretical tools, analytic techniques, degree counting method and sum of degrees of neighbours method [26,27]. Moreover, we use Matlab (MathWorks, Natick, MA, USA) for mathematical calculations and verifications (see https://en.wikipedia.org/wiki/MATLAB). We also use the maple software (Maplesoft, McKinney, TX, USA) for plotting these mathematical results (see https://en.wikipedia.org/wiki/Maple_(software)).

4. Oxide Network

Oxide networks play a vital role in the study of silicate networks. If we delete silicon vertices from a silicate network, we get an oxide network $O{X}_n$ (see Figure 1). An n-dimensional oxide network is denoted as $O{X}_n$. The number of vertices in Oxide network are $9n^2+3n$ and number of edges are $18n^2$. An Oxide network $O{X}_n$ with $n=5$ is depicted in Figure 1.

4.1. Construction of Oxide Network $O{X}_n$ Formulas

• To prove our main results, we make a partition of vertices of the oxide network $O{X}_n$ for (n-levels) based on eccentricity of each vertex in two sets. The set $V_1$ contains those vertices which have the eccentricity $\epsilon \left(u\right)=2k+1$, and the number of vertices in set $V_1$ are $6(2m-1)$, $1\le m\le n$. The set $V_2$ contain those vertices which have the eccentricity $\epsilon \left(u\right)=2k+2$, and the number of vertices in set $V_2$ are $6m$, $1\le m\le n$. Also, the variable k represents the distance between two vertices, which helps us to make this vertex partition. Also, k represents the range of the total number of vertices with that eccentricity. More preciously,

  Table 1. Vertex partition of oxide network for (n-levels) based on eccentricity of each vertex.

  $\mathit{\epsilon }\left(\mathit{u}\right)$	Number of Vertices	Range of k	Range of m and n	Sets
  $2k+1$	$6(2m-1)$	$n\le k\le 2n-1$	$1\le m\le n$, $n\ge 1$	$V_1$
  $2k+2$	$6m$	$n\le k\le 2n-1$	$1\le m\le n$, $n\ge 1$	$V_2$

  represents the vertex partition of Oxide network for (n-levels) based on eccentricity of each vertex.
• Now we make a partition of edges of an oxide network for (n-levels) based on eccentricity of end vertices in three sets. The set $E_1$ contain those edges which have the eccentricities $\left(\epsilon \right(u),\epsilon (v\left)\right)=(2k+1,2k+1)$, $n\le k\le 2n-1$ and the number of edges in set $E_1$ are $6(2m-1)$, $1\le m\le n$. The set $E_2$ contain those edges which have the eccentricities $\left(\epsilon \right(u),\epsilon (v\left)\right)=(2k+1,2k+2)$, $n\le k\le 2n-1$, and the number of edges in set $E_2$ are $12m$, $1\le m\le n$. The set $E_3$ contain those edges which have the eccentricities $\left(\epsilon \right(u),\epsilon (v\left)\right)=(2k+2,2k+3)$, $n\le k\le 2n-1$, and the number of edges in set $E_3$ are $12m$, $1\le m\le n$. Also k represent the range of total number of pairs with that eccentricity. More preciously

  Table 2. Edge partition of oxide network for (n-levels) based on eccentricity of end vertices.

  $\left(\mathit{\epsilon }\right(\mathit{u}),\mathit{\epsilon }(\mathit{v}\left)\right)$	Number of Edges	Range of k	Range of m and n	Sets
  $(2k+1,2k+1)$	$6(2m-1)$	$n\le k\le 2n-1$	$1\le m\le n$, $n\ge 1$	$E_1$
  $(2k+1,2k+2)$	$12m$	$n\le k\le 2n-1$	$1\le m\le n$, $n\ge 1$	$E_2$
  $(2k+2,2k+3)$	$12m$	$n\le k\le 2n-2$	$1\le m\le n-1$, $n>1$	$E_3$

  represents the edge partition of oxide network for (n-levels) based on eccentricity of end vertices.

4.2. Main Results for Oxide Network

In this section, we computed the close formulae of certain topological indices for this network. Here we find the analytically closed results of total eccentricity index, average eccentricity index, eccentricity based Zagreb indices, eccentricity based geometric arithmetic and atom-bond connectivity indices for oxide networks.

Theorem 1.

Let $O{X}_n$, for all $n\in N$, be the oxide network, then the total eccentricity index ζ of $O{X}_n$ is

$$\zeta \left(O{X}_n\right)=6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{6mk+4m-2k-1\}.$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

Using the vertex partitioned from Table 1 and Equation (2), we have computed the total eccentricity index as:

$$\begin{array}{ccc}\hfill \zeta \left(G\right)& =& \sum _{v\in V\left(G\right)}\epsilon \left(v\right)\hfill \\ \hfill \zeta \left(O{X}_n\right)& =& \sum _{v\in V_1\left(G\right)}\epsilon \left(v\right)+\sum _{v\in V_2\left(G\right)}\epsilon \left(v\right)\hfill \\ & =& \sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)·(2k+1)+\sum _{m=1}^n\sum _{k=n}^{2n-1}6m·(2k+2)\hfill \\ & =& 6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{(2m-1)·(2k+1)+m·(2k+2)\}\hfill \end{array}$$

After an easy simplification, we get

$$\zeta \left(O{X}_n\right)=6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{6mk+4m-2k-1\}.$$

□

Theorem 2.

Let $O{X}_n$, for all $n\in N$, be the oxide network, then the average eccentricity index $avec$ of $O{X}_n$ is

$$avec\left(O{X}_n\right)=\frac{2}{3n^2+n}\sum _{m=1}^n\sum _{k=n}^{2n-1}\{6mk+4m-2k-1\}.$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

Using the vertex partitioned from Table 1 and Equation (3), we have computed the average eccentricity index of oxide network $avec\left(O{X}_n\right)$ as:

$$\begin{array}{ccc}\hfill avec\left(G\right)& =& \frac{1}{n}\sum _{v\in V\left(G\right)}\epsilon \left(v\right)\hfill \\ \hfill avec\left(O{X}_n\right)& =& \frac{1}{n}\sum _{v\in V_1\left(G\right)}\epsilon \left(v\right)+\frac{1}{n}\sum _{v\in V_2\left(G\right)}\epsilon \left(v\right)\hfill \\ \hfill avec\left(O{X}_n\right)& =& \frac{1}{9n^2+3n}\{\sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)·(2k+1)+\sum _{m=1}^n\sum _{k=n}^{2n-1}6m·(2k+2)\}\hfill \end{array}$$

After an easy simplification, we get

$$avec\left(O{X}_n\right)=\frac{2}{3n^2+n}\sum _{m=1}^n\sum _{k=n}^{2n-1}\{6mk+4m-2k-1\}.$$

□

Theorem 3.

Let $O{X}_n$ for all $n\in N$, be the oxide network, then the first Zagreb eccentricity index $M_1^{*}\left(O{X}_n\right)$ is

$$M_1^{*}\left(O{X}_n\right)=12\sum _{m=1}^n\sum _{k=n}^{2n-1}\{8mk+5m-2k-1\}+12\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m(4k+5).$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

Using the vertex partitioned from Table 2 and Equation (6), we have computed first Zagreb eccentricity index $M_1^{*}\left(O{X}_n\right)$ as:

$$\begin{array}{ccc}\hfill M_1^{*}\left(G\right)& =& \sum _{uv\in E\left(G\right)}[\epsilon \left(u\right)+\epsilon \left(v\right)]\hfill \\ \hfill M_1^{*}\left(O{X}_n\right)& =& \sum _{uv\in E_1\left(G\right)}[\epsilon \left(u\right)+\epsilon \left(v\right)]+\sum _{uv\in E_2\left(G\right)}[\epsilon \left(u\right)+\epsilon \left(v\right)]+\sum _{uv\in E_3\left(G\right)}[\epsilon \left(u\right)+\epsilon \left(v\right)]\hfill \\ & =& \sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)(2k+1+2k+1)+\sum _{m=1}^n\sum _{k=n}^{2n-2}12m(2k+1+2k+2)\hfill \\ & +& \sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}12m(2k+2+2k+3)\hfill \\ & =& 6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{(2m-1)(4k+2)+2m(4k+3)\}+12\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m(4k+5).\hfill \end{array}$$

After some simplification, we obtain

$$M_1^{*}\left(O{X}_n\right)=12\sum _{m=1}^n\sum _{k=n}^{2n-1}\{8mk+5m-2k-1\}+12\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m(4k+5).$$

□

Theorem 4.

Let $O{X}_n$ for all $n\in N$, be the oxide network, then the second Zagreb eccentricity index $M_1^{**}\left(O{X}_n\right)$ is

$$M_1^{**}\left(O{X}_n\right)=6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{2m(6k^2+8k+3)-{(2k+1)}^2\}.$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

The general formula of second Zagreb eccentricity index is

$$M_1^{**}\left(G\right)=\sum _{v\in V\left(G\right)}{\left[\epsilon \left(v\right)\right]}^2.$$

$$M_1^{**}\left(G\right)=\sum _{v\in V_1\left(G\right)}{\left[\epsilon \left(v\right)\right]}^2+\sum _{v\in V_2\left(G\right)}{\left[\epsilon \left(v\right)\right]}^2.$$

By using the values from Table 1 , we have

$$M_1^{**}\left(O{X}_n\right)=\sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)·{(2k+1)}^2+\sum _{m=1}^n\sum _{k=n}^{2n-1}6m·{(2k+2)}^2.$$

$$M_1^{**}\left(O{X}_n\right)=6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{(2m-1)·{(2k+1)}^2+4m·{(k+1)}^2\}.$$

After some simplification, we obtain

$$M_1^{**}\left(O{X}_n\right)=6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{2m(6k^2+8k+3)-{(2k+1)}^2\}.$$

□

Theorem 5.

Let $O{X}_n$ for all $n\in N$, be the oxide network, then the third Zagreb eccentricity index $M_2^{*}\left(O{X}_n\right)$ is

$$M_2^{*}\left(O{X}_n\right)=12\sum _{m=1}^n\sum _{k=n}^{2n-1}\{2m(8k^2+8k+3)-(4k^2+2k+1)\}+24\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m(2k+3)(k+1).$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

The general formula of third Zagreb eccentricity index is

$$M_2^{*}\left(G\right)=\sum _{uv\in E\left(G\right)}[\epsilon \left(u\right)·\epsilon \left(v\right)].$$

$$M_2^{*}\left(G\right)=\sum _{uv\in E_1\left(G\right)}[\epsilon \left(u\right)·\epsilon \left(v\right)]+\sum _{uv\in E_2\left(G\right)}[\epsilon \left(u\right)·\epsilon \left(v\right)]+\sum _{uv\in E_3\left(G\right)}[\epsilon \left(u\right)·\epsilon \left(v\right)]$$

By using the values from Table 2, we have

$$\begin{array}{ccc}\hfill M_2^{*}\left(O{X}_n\right)& =& \sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)(2k+1)·(2k+1)+\sum _{m=1}^n\sum _{k=n}^{2n-1}12m(2k+1)·(2k+2)\hfill \\ & +& \sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}12m(2k+2)·(2k+3).\hfill \end{array}$$

$$M_2^{*}\left(O{X}_n\right)=6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{(2m-1){(2k+1)}^2+2m(4k^2+4k+2k+2)\}+24\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m(k+1)(2k+3).$$

After some simplification, we obtain

$$M_2^{*}\left(O{X}_n\right)=12\sum _{m=1}^n\sum _{k=n}^{2n-1}\{2m(8k^2+8k+3)-(4k^2+2k+1)\}+24\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m(2k+3)(k+1).$$

□

Theorem 6.

Let $O{X}_n$ for all $n\in N$, be the oxide network, then the geometric-arithmetic index $G{A}_4\left(O{X}_n\right)$ is

$$\begin{array}{ccc}\hfill G{A}_4\left(O{X}_n\right)& =& 12\sum _{m=1}^n\sum _{k=n}^{2n-1}\left\{\frac{2m-1}{2}+2m\frac{\sqrt{(2k+1)(2k+2)}}{(4k+3)}\right\}\hfill \\ & +& 24\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m\sqrt{\frac{(2k+2)(2k+3)}{4k+5}}.\hfill \end{array}$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

The general formula of eccentricity based geometric arithmetic index is

$$G{A}_4\left(G\right)=\sum _{uvϵE\left(G\right)}\frac{2\sqrt{\epsilon \left(u\right)·\epsilon \left(v\right)}}{\epsilon \left(u\right)+\epsilon \left(v\right)}.$$

$$G{A}_4\left(G\right)=\sum _{uvϵE_1\left(G\right)}\frac{2\sqrt{\epsilon \left(u\right)·\epsilon \left(v\right)}}{\epsilon \left(u\right)+\epsilon \left(v\right)}+\sum _{uvϵE_2\left(G\right)}\frac{2\sqrt{\epsilon \left(u\right)·\epsilon \left(v\right)}}{\epsilon \left(u\right)+\epsilon \left(v\right)}+\sum _{uvϵE_3\left(G\right)}\frac{2\sqrt{\epsilon \left(u\right)·\epsilon \left(v\right)}}{\epsilon \left(u\right)+\epsilon \left(v\right)}$$

Using the edge partitioned from Table 2, we have the following computations

$$\begin{array}{ccc}\hfill G{A}_4\left(OXn\right)& =& \sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)·2\frac{\sqrt{(2k+1)·(2k+1)}}{2k+1+2k+1}+\sum _{m=1}^n\sum _{k=n}^{2n-1}12m·2\frac{\sqrt{(2k+1)·(2k+2)}}{2k+1+2k+2}\hfill \\ & +& \sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}12m·2\frac{\sqrt{(2k+2)·(2k+3)}}{2k+2+2k+3}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill G{A}_4\left(OXn\right)& =& 12\sum _{m=1}^n\sum _{k=n}^{2n-1}\left\{(2m-1)\frac{\sqrt{{(2k+1)}^2}}{4k+2}+2\frac{\sqrt{(2k+1)·(2k+2)}}{4k+3}\right\}\hfill \\ & +& 24\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m\frac{\sqrt{(2k+2)·(2k+3)}}{4k+5}.\hfill \end{array}$$

After some simplification, we obtain

$$\begin{array}{ccc}\hfill G{A}_4\left(O{X}_n\right)& =& 12\sum _{m=1}^n\sum _{k=n}^{2n-1}\left\{\frac{2m-1}{2}+2m\frac{\sqrt{(2k+1)(2k+2)}}{(4k+3)}\right\}\hfill \\ & +& 24\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m\sqrt{\frac{(2k+2)(2k+3)}{4k+5}}.\hfill \end{array}$$

□

Theorem 7.

Let $O{X}_n$ for all $n\in N$, be the oxide network, then the atom-bond connectivity index $AB{C}_5\left(O{X}_n\right)$ is

$$\begin{array}{ccc}\hfill AB{C}_5\left(O{X}_n\right)& =& 12\sum _{m=1}^n\sum _{k=n}^{2n-1}\left\{\frac{(2m-1)\sqrt{k}}{2k+1}+m\sqrt{\frac{4k+1}{(2k+1)(2k+2)}}\right\}\hfill \\ & +& 12\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m\sqrt{\frac{4k+3}{(2k+2)(2k+3)}}.\hfill \end{array}$$

Proof. 

Let $O{X}_n$, where $n\in N$, be the oxide network containing $9n^2+3n$ vertices and $18n^2$ edges.

The general formula of eccentricity based atom-bond connectivity index is

$$AB{C}_5\left(G\right)=\sum _{uvϵE\left(G\right)}\sqrt{\frac{\epsilon \left(u\right)+\epsilon \left(v\right)-2}{\epsilon \left(u\right)·\epsilon \left(v\right)}}.$$

$$AB{C}_5\left(G\right)=\sum _{uvϵE_1\left(G\right)}\sqrt{\frac{\epsilon \left(u\right)+\epsilon \left(v\right)-2}{\epsilon \left(u\right)·\epsilon \left(v\right)}}+\sum _{uvϵE_2\left(G\right)}\sqrt{\frac{\epsilon \left(u\right)+\epsilon \left(v\right)-2}{\epsilon \left(u\right)·\epsilon \left(v\right)}}+\sum _{uvϵE_3\left(G\right)}\sqrt{\frac{\epsilon \left(u\right)+\epsilon \left(v\right)-2}{\epsilon \left(u\right)·\epsilon \left(v\right)}}.$$

Using the edge partitioned from Table 2, we have the following computations

$$\begin{array}{ccc}\hfill AB{C}_5\left(O{X}_n\right)& =& \sum _{m=1}^n\sum _{k=n}^{2n-1}6(2m-1)\sqrt{\frac{2k+1+2k+1-2}{(2k+1)·(2k+1)}}+\sum _{m=1}^n\sum _{k=n}^{2n-1}12m\sqrt{\frac{2k+1+2k+2-2}{(2k+1)·(2k+2)}}\hfill \\ & +& \sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}12m\sqrt{\frac{2k+2+2k+3-2}{(2k+2)·(2k+3)}}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill AB{C}_5\left(O{X}_n\right)& =& 6\sum _{m=1}^n\sum _{k=n}^{2n-1}\{(2m-1)\sqrt{\frac{4k}{{(2k+1)}^2}}+2m\sqrt{\frac{4k+1}{(2k+1)·(2k+2)}}\}\hfill \\ & +& 12\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m\sqrt{\frac{4k+3}{(2k+2)·(2k+3)}}.\hfill \end{array}$$

After some simplification, we obtain

$$\begin{array}{ccc}\hfill AB{C}_5\left(O{X}_n\right)& =& 12\sum _{m=1}^n\sum _{k=n}^{2n-1}\left\{\frac{(2m-1)\sqrt{k}}{2k+1}+m\sqrt{\frac{4k+1}{(2k+1)(2k+2)}}\right\}\hfill \\ & +& 12\sum _{m=1}^{n-1}\sum _{k=n}^{2n-2}m\sqrt{\frac{4k+3}{(2k+2)(2k+3)}}.\hfill \end{array}$$

□

5. Comparisons and Discussion

For the comparison of these indices numerically for $O{X}_n$, we computed all indices for different values of $m,k$. Now, from

Table 3. Numerical computation of all indices for $O{X}_n$.

$[\mathit{m},\mathit{k}]$	$\mathit{\zeta }\left(\mathit{G}\right)$	$\mathbf{avec}\left(\mathit{G}\right)$	${\mathit{M}}_1^{*}\left(\mathit{G}\right)$	${\mathit{M}}_1^{**}\left(\mathit{G}\right)$	${\mathit{M}}_2^{*}\left(\mathit{G}\right)$	${\mathbf{GA}}_4\left(\mathit{G}\right)$	${\mathbf{ABC}}_5\left(\mathit{G}\right)$
$[1,1]$	42	$1.9$	1416	2568	2014	$112.5$	$315.4$
$[2,2]$	162	$3.5$	4188	5478	4352	$279.9$	$645.3$
$[3,3]$	354	$5.6$	8304	10,523	9300	$446.7$	$987.4$
$[4,4]$	618	$8.4$	13,764	14,587	11,248	$613.6$	$1125.6$
$[5,5]$	956	$10.5$	16,898	16,325	13,654	$842.3$	$1356.4$
$[6,6]$	1242	$14.5$	19,652	19,876	16,324	$1023.3$	$1586.7$

, we can easily see that all indices are in increasing order as the values of $m,k$ are increasing. The graphical representations of the all indices for $O{X}_n$ are depicted in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 for certain values of $m,k$.

6. Conclusions

In this paper, we computed the total eccentricity index $\zeta \left(O{X}_n\right)$, average eccentricity index $avec\left(O{X}_n\right)$, eccentricity-based Zagreb indices $M_1^{*}\left(O{X}_n\right)$, $M_1^{**}\left(O{X}_n\right)$ and $M_2^{*}\left(O{X}_n\right)$, atom-bond connectivity index $AB{C}_5\left(O{X}_n\right)$ and geometric arithmetic index $G{A}_4\left(O{X}_n\right)$ of the oxide network $O{X}_n$. So these indices are useful to analyzed the physico-chemical, pharmacological and toxicological properties of the oxide network $O{X}_n$.