Comparative Study of Prism Octahedron Network via Eccentric Invariants

Abstract

Topological indices are empirical features of graphs that characterize the topology of the graph and, for the most part, are graph independent. An important branch of graph theory is chemical graph theory. In chemical graph theory, the atoms corresponds vertices and edges corresponds covalent bonds. A topological index is a numeric number that represents the topology of underline structure. In this article, we examined the topological properties of prism octahedron network of dimension $m$ and computed the total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to utilize the distance between the vertices of a prism octahedron network.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminary Results

Cheminformatics [1] is a new field of study in graph theory recent time. Physicists define a modern scientific discipline that blends chemistry, mathematics, and computer science [2]. Quantitative structure activity relationships (QSAR) and quantitative structure property relationships (QSPR) are frequently used to predict molecular biological activities and properties [3]. That is why, they have aroused the curiosity of scholars all around the world because they are used in quantitative nonempirical structures, property connections, quantitative structure activity correlations, and elements of topology, which are significant in the inquiry of computational chemistry. Topological descriptor $\mathrm{T}\mathrm{o}\mathrm{p}G$ can also be defined in terms of isomorphisms. $\mathrm{T}\mathrm{o}\mathrm{p}G=\mathrm{T}\mathrm{o}\mathrm{p}H$, for every isomorphic graph $H$ to $G$. Wiener [4] first developed the notion of topological indices in 1947, during working in the laboratory on the critical temperature index while working in the laboratory on the melting point of paraffin. In this article, we investigate how a prism octahedron network of dimension $m$ may be constructed from a silicon oxide structure [5] by swapping each silicon node into $K_3$, a complete graph with three vertices, and we determine the eccentricity of the prism octahedron network.

A graph $G=V,E$, where $V$ and $E$ are nonempty sets of vertices and edges, respectively. Mathematically, graph theory is utilised in chemical graph theory to represent molecular [6] processes, which are beneficial for researching and applying a diverse set of topological indices. There are two varieties of chemical graph theory in theoretical chemistry. Numerous topological indices for graphs are significant in the advancement of chemical scientific investigation. $Dr,s$ denotes the distance between $r$ and $s$ and is defined as the length of the shortest path in $G$ if $r,s\in \mathbb{V}G$.

If $r\in VG$, eccentricity is defined as the maximum distance between a vertex r to all other vertices in the graph.

In terms of numbers$$\begin{array}{}\text{(1)}& \mathrm{\varrho }b=\max dr,sb\in VG\text{.}\end{array}$$

The total eccentricity [7] index is defined as$$\begin{array}{}\text{(2)}& \vartheta G={\displaystyle \sum }_{b\in VG}\mathrm{\varrho }b\text{,}\end{array}$$where $\mathrm{\varrho }b$ is the eccentricity of the vertex $b$.

The graph average eccentricity avec $\widehat{G}$ [8] index is defined as$$\begin{array}{}\text{(3)}& \mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}\widehat{G}=\frac{1}{\mathcal{I}}{\displaystyle \sum }_{b\in VG}\mathrm{\varrho }b\text{,}\end{array}$$where for the $m$ dimension of a graph, $\mathcal{I}$ denotes the total number of vertices. Many academics, including Ilic and Tang, have engaged in the average eccentricity index [8, 9].

The eccentricity geometric arithmetic index [10, 11] is as follows:$$\begin{array}{}\text{(4)}& {\mathrm{G}\mathrm{A}}_{4}G={\displaystyle \sum }_{ub\in \mathrm{\varrho }G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}\text{.}\end{array}$$

The eccentricity version of the ABC index is formulated as follows:$$\begin{array}{}\text{(5)}& {\mathrm{A}\mathrm{B}\mathrm{C}}_{5}G={\displaystyle \sum }_{ub\in \mathrm{\varrho }G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}\text{.}\end{array}$$

The first and second Zagreb eccentricity indexes are as follows:$$\begin{array}{c}\text{(6)}& {\varpi }_{1}G={\displaystyle \sum }_{b\in VG}{\mathrm{\varrho }b}^2\text{,}{\varpi }_{2}G={\displaystyle \sum }_{ub\in \mathrm{\varrho }G}\mathrm{\varrho }u\mathrm{\varrho }b\text{.}\end{array}$$

2. Main Results

This section discusses the prism octahedron network $P_r\mathrm{O}\mathrm{H}m$ with $m$ dimension as shown in Figure 1. The closed analytical finding for the total and average eccentricity index [12, 13], eccentricity index [14], eccentricity-based Zagreb indices [15, 16], geometric arithmetic [17, 18], and atom bond connectivity is computed.

[figure(s) omitted; refer to PDF]

The order and size of $P_r\mathrm{O}\mathrm{H}m$ may be calculated using a simple computation, $i.e\text{.}$, $V{P}_r\mathrm{O}\mathrm{H}m$ = $27m^2+3m$ and $E{P}_r\mathrm{O}\mathrm{H}m$ = $54{\mathrm{m}}^2$, where $m$ is the graph’s dimension.

2.1. Results on Prism Octahedron Network

In chemistry, prism octahedron networks made of honeycomb structures [19, 20] are critical for researching polymers with low density and high bending. These frameworks are also employed to study stress in other aerospace-related materials. In this article, we look into the particular eccentricity of prism octahedron network. The vertex partition of ${\mathrm{P}}_{\mathrm{r}}\mathrm{O}\mathrm{H}m$ is shown in Table 1.

Table 1

Vertex partition of ${\mathrm{P}}_{\mathrm{r}}\mathrm{O}\mathrm{H}m$.

Sets	$\mathrm{\varrho }b$	Vertices	Ranges
$V_1$	$2m+2b+6$	$24b+54$	$0\le b\le m-3$
$V_2$	$2m+b+2$	$12b+6$	$0\le b\le 2$
$V_3$	$2n+2b+5$	$30b+48$	$0\le b\le m-2$
$V_4$	$4m+2$	$6m$

Theorem 1.

For $P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$, as the graph of prism octahedron network, the total eccentricity $\phi$ index is equal to$$\begin{array}{}\text{(7)}& \phi P_r\mathrm{O}\mathrm{H}m=90m^3+57m^2+3m\text{.}\end{array}$$

Proof.

$P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$ are the prism octahedron network. Using the vertex partition from the Table 1, we compute the total eccentricity index as follows:$$\begin{array}{c}\text{(8)}& \phi G={\displaystyle \sum }_{b\in VG}\mathrm{\varrho }b\text{,}\phi P_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b\in v_{1}G}\mathrm{\varrho }b+{\displaystyle \sum }_{b\in v_{2}G}\mathrm{\varrho }b+{\displaystyle \sum }_{b\in v_{3}G}\mathrm{\varrho }b+\mathrm{\varrho }{b}_4\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b=0}^{m-3}2m+2b+624b+54+{\displaystyle \sum }_{b=0}^{2}2m+b+212b+6\\ +{\displaystyle \sum }_{b=0}^{m-2}2m+2b+530b+48+4m+26m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=24m{m-2}^2+16{m-2}^3+84mm-2\\ +102{m-2}^2+314m-226\\ +{\displaystyle \sum }_{b=0}^{m-2}2m+2b+530b+48+4m+26m\text{,}\\ P_r\mathrm{O}\mathrm{H}m=24m{m-2}^2+16{m-2}^3+84mm-2\\ +102{m-2}^2+441m-353+30m{m-1}^2\\ +20{m-1}^3+66mm-1+93{m-1}^2+4m+26m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=24m^3-96m^2+96m+16m^3-96m^2+192m-128\\ +84m^2-168m+102m^2-408m+408+441m-353\\ +30m^3-60m^2+30m+20m^3-60m^2+60m-20\\ +66m^2-66m+93m^2-186m+93+24m^2+12m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=24m^3+16m^3+30m^3+20m^3-96m^2-96m^2+84m^2\\ +102m^2-60m^2-60m^2+66m^2+93m^2+24m^2\\ +96m+192m-168m-408m+441m+30m+60m\\ -66m-186m+12m-128+408-353-20+93\text{.}\end{array}$$

After calculation, we have$$\begin{array}{}\text{(9)}& \Rightarrow \varphi P_r\mathrm{O}\mathrm{H}m=90m^3+57m^2+3m\text{.}\end{array}$$

Theorem 2.

For $P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$, the graph of prism octahedron network, the avec index is equal to$$\begin{array}{}\text{(10)}& \mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}{P}_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}90m^3+57m^2+3m\text{.}\end{array}$$

Proof.

$P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$ include $27m^2+3m$ vertices and $54m^2$ edges. Using the vertices partition from Table 1, we obtained the average eccentricity index as follows:$$\begin{array}{c}\text{(11)}& \mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}\widehat{G}=\frac{1}{\mathcal{I}}{\displaystyle \sum }_{b\in VG}\mathrm{\varrho }b\text{,}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}{P}_r\mathrm{O}\mathrm{H}m=\frac{1}{\mathcal{I}}{\displaystyle \sum }_{b\in v_{1}G}\mathrm{\varrho }b+\frac{1}{\mathcal{I}}{\displaystyle \sum }_{b\in v_{2}G}\mathrm{\varrho }b+\frac{1}{\mathcal{I}}{\displaystyle \sum }_{b\in v_{3}G}\mathrm{\varrho }b+\frac{1}{\mathcal{I}}\mathrm{\varrho }{b}_4\text{,}\\ \mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}{P}_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}{\displaystyle \sum }_{b=0}^{m-3}2m+2b+624b+54+{\displaystyle \sum }_{b=0}^{2}2m+b+2\\ 12b+6+{\displaystyle \sum }_{b=0}^{m-2}2m+2b+530b+48+4m+26m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}24m{m-2}^2+16{m-2}^3+84mm-2\\ +102{m-2}^2+314m-226\\ +{\displaystyle \sum }_{b=0}^{m-2}2m+2b+530b+48+4m+26m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}24m{m-2}^2+16{m-2}^3+84mm-2\\ +102{m-2}^2+441m-353+30m{m-1}^2\\ +20{m-1}^3+66mm-1+93{m-1}^2+4m+26m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}24m^3-96m^2+96m16m^3-96m^2+192m-128\\ +84m^2-168m+102m^2-408m+408+441m-353\\ +30m^3-60m^2+30m+20m^3-60m^2+60m-20\\ +66m^2-66m+93m^2-186m+93+24m^2+12m\text{,}\\ \phi P_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}24m^3+16m^3+30m^3+20m^3-96m^2-96m^2+84m^2\\ +102m^2-60m^2-60m^2+66m^2+93m^2+24m^2\\ +96m+192m-168m-408m+441m+30m+60m\\ -66m-186m+12m-128+408-353-20+93\text{.}\end{array}$$

After calculation, we have$$\begin{array}{}\text{(12)}& \mathrm{a}\mathrm{v}\mathrm{e}\mathrm{c}{P}_r\mathrm{O}\mathrm{H}m=\frac{1}{27m^2+3m}90m^3+57m^2+3m\text{.}\end{array}$$

Theorem 3.

For $P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$, the graph of prism octahedron network, the Zagreb eccentricity index is equal to$$\begin{array}{}\text{(13)}& {\varpi }_1{P}_r\mathrm{O}\mathrm{H}m=306m^4+348m^3+555m^2+3m\text{.}\end{array}$$

Proof.

$P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$ and $m\ge 2$ are the prism octahedron network. Using the partitioned vertices from Table 1, we estimated the Zagreb eccentricity index as follows:$$\begin{array}{c}\text{(14)}& {\varpi }_{1}G={\displaystyle \sum }_{b\in VG}{\mathrm{\varrho }b}^2\text{,}{\varpi }_1{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b\in v_{1}G}{\mathrm{\varrho }b}^2+{\displaystyle \sum }_{b\in v_{2}G}{\mathrm{\varrho }b}^2+{\displaystyle \sum }_{b\in v_{3}G}{\mathrm{\varrho }b}^2+\mathrm{\varrho }{b_4}^2\text{,}\\ {\varpi }_1{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b=0}^{m-3}{2m+2b+6}^{2}24b+54+{\displaystyle \sum }_{b=0}^2{2m+b+2}^{2}12b+6\\ +{\displaystyle \sum }_{b=0}^{m-2}{2m+2b+5}^{2}30b+48+{4m+2}^{2}6m\text{,}\\ {\varpi }_1{P}_r\mathrm{O}\mathrm{H}m=48m^2{m-2}^2+64m{m-2}^3+24{m-2}^4\\ +168m^{4}m-2+408m{m-2}^2+216{m-2}^3\\ +824mm-2+708{m-2}^2+1473m-2469\\ +6{2m+2}^2+18{2m+3}^2+30{2m+4}^2\\ +60m^2{-1+m}^2+80m{-1+m}^3+30{-1+m}^4\\ +132m^2-1+m+372n{-1+m}^2+204{-1+m}^3\\ +508m-1+m+489{-1+m}^2+{4m+2}^{2}6m\text{,}\\ {\varpi }_1{P}_r\mathrm{O}\mathrm{H}m=48m^4-192m^3+192m^2+64m^4-384m^3\\ +768m^2-512m+24m^4-192m^3+576m^2\\ -768m+384+168m^3-336m^2+408m^3\\ -1632m^2+1632m+216m^3-1296m^2\\ +2592m-1728+824m^2-1648m\\ +708m^2-2832m+2832+1473m-2469\\ +24m^2+48m+24+72m^2+216m+162\\ +120m^2+480m+480+60m^4-120m^3\\ +60m^2+80m^4-240m^3+240m^2-80m\\ +30m^4-120m^3+180m^2-120m+30\\ +132m^3-132m^2+372m^3-744m^2+372m\\ +204m^3-162m^2+612m-204+508m^2-508m\\ +489m^2-978m+489+96m^3+96m^2+24m\text{,}\\ {\varpi }_1{P}_r\mathrm{O}\mathrm{H}m=48m^4+64m^4+24m^4+60m^4+80m^4+30m^4\\ -192m^3-384m^3-192m^3+168m^3+408m^3\\ +216m^3-120m^3-240m^3-120m^3+132m^3\\ +372m^3+204m^3+96m^3+192m^2+768m^2\\ +576m^2-336m^2-1632m^2-1296m^2+824m^2\\ +708m^2+24m^2+72m^2+120m^2+60m^2+240m^2\\ +180m^2-132m^2-744m^2-162m^2+508m^2+489m^2\\ +96m^2-512m-768m+1632m+2592m-1648m-2832m\\ +1473m+48m+216m+480m-80m-120m+372m\\ +612m-508m-978m+24m+384-1728+2832\\ -2469+24+162+480+30-204+489\text{.}\end{array}$$

After calculation, we have$$\begin{array}{}\text{(15)}& {\varpi }_1{P}_r\mathrm{O}\mathrm{H}m=306m^4+348m^3+555m^2+3m\text{.}\end{array}$$

Theorem 4.

For $P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$,and $m\ge 2$, represent the graph of the prism octahedron network, the ${\mathrm{G}\mathrm{A}}_4$ index has to be equal to$$\begin{array}{c}\text{(16)}& {\mathrm{G}\mathrm{A}}_4{P}_r\mathrm{O}\mathrm{H}m=72m+48{\displaystyle \sum }_{b=0}^{m-1}b\frac{2\sqrt{2m2m+2b+2+2b2m+2b+2+3}}{4m+4b+5}-36+36{\displaystyle \sum }_{b=0}^{m-2}b\frac{2\sqrt{2b2m+2b+3+2m2m+2b+3+4}}{4m+4b+7}\\ +12{\displaystyle \sum }_{b=0}^{m-1}b\frac{\sqrt{2m2m+2b+3+2b2m+2b+3+3}}{2m+2b+3}\\ +12{\displaystyle \sum }_{b=0}^{m-1}b\frac{\sqrt{2m2m+2b+2+2b2m+2b+2+2}}{2m+2b+2}\\ +18m^2\frac{2\sqrt{4m4m+2+1}}{8m+3}\text{.}\end{array}$$

Proof.

$P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$. Using Table 2 edge partition, we arrive at the eccentric ${\mathrm{G}\mathrm{A}}_4$ index as follows:$$\begin{array}{c}\text{(17)}& {\mathrm{G}\mathrm{A}}_{4}G={\displaystyle \sum }_{ub\in \mathrm{\varrho }G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}\text{,}{GA}_4{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{1}G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}+{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{2}G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}\\ +{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{3}G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}+{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{4}G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}\\ +{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{5}G}\frac{2\sqrt{\mathrm{\varrho }u.\mathrm{\varrho }b}}{\mathrm{\varrho }u+\mathrm{\varrho }b}\text{,}\\ {\mathrm{G}\mathrm{A}}_4{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b=0}^{m-1}48b+24\frac{2\sqrt{2m+2b+3.2m+2b+2}}{2m+2b+3+2m+2b+2}\\ +{\displaystyle \sum }_{b=0}^{m-2}36b+36\frac{2\sqrt{2m+2b+4.2m+2b+3}}{2m+2b+4+2m+2b+3}\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+6\frac{2\sqrt{2m+2b+3.2m+2b+3}}{2m+2b+3+2m+2b+3}\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+6\frac{2\sqrt{2m+2b+2.2m+2b+2}}{2m+2b+2+2m+2b+2}\\ +18m\frac{2\sqrt{4m+1.4m+2}}{4m+1+4m+2}\text{,}\\ {\mathrm{G}\mathrm{A}}_4{P}_r\mathrm{O}\mathrm{H}m=24{\displaystyle \sum }_{b=0}^{m-1}2b+1\frac{2\sqrt{2m+2b+3.2m+2b+2}}{2m+2b+3+2m+2b+2}\\ +36{\displaystyle \sum }_{b=0}^{m-2}b+1\frac{2\sqrt{2m+2b+4.2m+2b+3}}{2m+2b+4+2m+2b+3}\\ +6{\displaystyle \sum }_{b=0}^{m-1}2b+1\frac{2\sqrt{2m+2b+3.2m+2b+3}}{2m+2b+3+2m+2b+3}\\ +6{\displaystyle \sum }_{b=0}^{m-1}2b+1\frac{2\sqrt{2m+2b+2.2m+2b+2}}{2m+2b+2+2m+2b+2}\\ +18m\frac{2\sqrt{4m+1.4m+2}}{4m+1+4m+2}\text{.}\end{array}$$

After calculation, we have$$\begin{array}{c}\text{(18)}& {\mathrm{G}\mathrm{A}}_4{P}_r\mathrm{O}\mathrm{H}m=72m+48{\displaystyle \sum }_{b=0}^{m-1}b\frac{2\sqrt{2m2m+2b+2+2b2m+2b+2+3}}{4m+4b+5}-36+36{\displaystyle \sum }_{b=0}^{m-2}b\frac{2\sqrt{2b2m+2b+3+2m2m+2b+3+4}}{4m+4b+7}\\ +12{\displaystyle \sum }_{b=0}^{m-1}b\frac{\sqrt{2m2m+2b+3+2b2m+2b+3+3}}{2m+2b+3}\\ +12{\displaystyle \sum }_{b=0}^{m-1}b\frac{\sqrt{2m2m+2b+2+2b2m+2b+2+2}}{2m+2b+2}\\ +18m^2\frac{2\sqrt{4m4m+2+1}}{8m+3}\text{.}\end{array}$$

Table 2

Edge partition of ${\mathrm{P}}_{\mathrm{r}}\mathrm{O}\mathrm{H}m$.

Sets	$\mathrm{\varrho }b,\mathrm{\varrho }u$	Edges	Ranges
$E_1$	$2m+2b+2,2m+2b+3$	$48b+24$	$0\le b\le m-1$
$E_2$	$2m+2b+3,2m+2b+4$	$36b+36$	$0\le b\le m-2$
$E_3$	$2m+2b+3,2m+2b+3$	$12b+6$	$0\le b\le m-1$
$E_4$	$2m+2b+2,2m+2b+2$	$12b+6$	$0\le b\le m-1$
$E_5$	$4m+1,4m+2$	$18m$

Theorem 5.

For $P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$, the graph of prism octahedron network, the ${\mathrm{A}\mathrm{B}\mathrm{C}}_5$ index is equal to$$\begin{array}{c}\text{(19)}& {\mathrm{A}\mathrm{B}\mathrm{C}}_5{P}_r\mathrm{O}\mathrm{H}m=72m+48{\displaystyle \sum }_{b=0}^{m-1}b\sqrt{\frac{2m2m+2b+2+2b2m+2b+2+1}{4m+4b+5}}-36+36{\displaystyle \sum }_{b=0}^{m-2}b\sqrt{2}\sqrt{\frac{m2m+2b+3+b2m+2b+3+1}{4m+4b+7}}\\ +6{\displaystyle \sum }_{b=0}^{m-1}b\sqrt{2}\sqrt{\frac{2m2m+2b+3+2b2m+2b+3+1}{2m+2b+3}}\\ +6{\displaystyle \sum }_{b=0}^{m-1}b\sqrt{2}\sqrt{\frac{m2m+2b+2+b2m+2b+2}{m+b+1}}\\ +18m^2\sqrt{\frac{4m4m+2-1}{8m+3}}\text{.}\end{array}$$

Proof.

$P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$ are the graph prism octahedron network. Applying Table 2 edge partition, we estimated the eccentric ${\mathrm{A}\mathrm{B}\mathrm{C}}_5$ index as follows:$$\begin{array}{c}\text{(20)}& {\mathrm{A}\mathrm{B}\mathrm{C}}_5{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{ub\in \mathrm{\varrho }G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}\text{,}{\mathrm{A}\mathrm{B}\mathrm{C}}_5{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{1}G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}+{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{2}G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}\\ +{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{3}G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}+{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{4}G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}\\ +{\displaystyle \sum }_{ub\in {\mathrm{\varrho }}_{5}G}\sqrt{\frac{\mathrm{\varrho }u+\mathrm{\varrho }b-2}{\mathrm{\varrho }u.\mathrm{\varrho }b}}\text{,}\\ {\mathrm{A}\mathrm{B}\mathrm{C}}_5{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b=0}^{m-1}48b+24\sqrt{\frac{2m+2b+2+2m+2b+3-2}{2m+2b+2.2m+2b+3}}\\ +{\displaystyle \sum }_{b=0}^{m-2}36b+36\sqrt{\frac{2m+2b+3+2m+2b+4-2}{2m+2b+3.2m+2b+4}}\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+6\sqrt{\frac{2m+2b+3+2m+2b+3-2}{2m+2b+3.2m+2b+3}}\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+6\sqrt{\frac{2m+2b+2+2m+2b+2-2}{2m+2b+2.2m+2b+2}}\\ +18m\sqrt{\frac{4m+1+4m+2-2}{4m+1.4m+2}}\text{,}\\ {\mathrm{A}\mathrm{B}\mathrm{C}}_5{P}_r\mathrm{O}\mathrm{H}m=24{\displaystyle \sum }_{b=0}^{m-1}2b+1\sqrt{\frac{2m+2b+2+2m+2b+3-2}{2m+2b+2.2m+2b+3}}\\ +36{\displaystyle \sum }_{b=0}^{m-2}b+1\sqrt{\frac{2m+2b+3+2m+2b+4-2}{2m+2b+3.2m+2b+4}}\\ +6{\displaystyle \sum }_{b=0}^{m-1}2b+1\sqrt{\frac{2m+2b+3+2m+2b+3-2}{2m+2b+3.2m+2b+3}}\\ +6{\displaystyle \sum }_{b=0}^{m-1}2b+1\sqrt{\frac{2m+2b+2+2m+2b+2-2}{2m+2b+2.2m+2b+2}}\\ +18m\sqrt{\frac{4m+1+4m+2-2}{4m+1.4m+2}}\text{.}\end{array}$$

After calculation, we have$$\begin{array}{c}\text{(21)}& {\mathrm{A}\mathrm{B}\mathrm{C}}_5{P}_r\mathrm{O}\mathrm{H}m=72m+48{\displaystyle \sum }_{b=0}^{m-1}b\sqrt{\frac{2m2m+2b+2+2b2m+2b+2+1}{4m+4b+5}}-36+36{\displaystyle \sum }_{b=0}^{m-2}b\sqrt{2}\sqrt{\frac{m2m+2b+3+b2m+2b+3+1}{4m+4b+7}}\\ +6{\displaystyle \sum }_{b=0}^{m-1}b\sqrt{2}\sqrt{\frac{2m2m+2b+3+2b2m+2b+3+1}{2m+2b+3}}\\ +6{\displaystyle \sum }_{b=0}^{m-1}b\sqrt{2}\sqrt{\frac{m2m+2b+2+b2m+2b+2}{m+b+1}}\\ +18m^2\sqrt{\frac{4m4m+2-1}{8m+3}}\text{.}\end{array}$$

Theorem 6.

For $P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$, the graph of prism octahedron network, the ${\varpi }_{2}G$ index is defined as follows:$$\begin{array}{}\text{(22)}& {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=612m^4+540m^3+114m^2\text{.}\end{array}$$

Proof.

$P_r\mathrm{O}\mathrm{H}m$, $\forall m\in \mathbb{N}$, and $m\ge 2$ are the prism octahedron network. Utilizing Table 2 edge splitting, we derived the third Zagreb eccentricity index as follows:$$\begin{array}{c}\text{(23)}& {\varpi }_{2}G={\displaystyle \sum }_{b\in VG}\mathrm{\varrho }u\mathrm{\varrho }b\text{,}{\varpi }_2{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b\in V_{1}G}\mathrm{\varrho }u\mathrm{\varrho }b+{\displaystyle \sum }_{b\in V_{2}G}\mathrm{\varrho }u\mathrm{\varrho }b\\ +{\displaystyle \sum }_{b\in V_{3}G}\mathrm{\varrho }u\mathrm{\varrho }b+{\displaystyle \sum }_{b\in V_{4}G}\mathrm{\varrho }u\mathrm{\varrho }b\\ +{\displaystyle \sum }_{b\in V_{5}G}\mathrm{\varrho }u\mathrm{\varrho }b\text{,}\\ {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m={\displaystyle \sum }_{b=0}^{m-1}48b+242m+2b+22m+2b+3\\ +{\displaystyle \sum }_{b=0}^{m-2}36b+362m+2b+32m+2b+4\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+62m+2b+32m+2b+3\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+62m+2b+22m+2b+2\\ +18m4m+14m+2\text{,}\\ {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=272m^4+240m^3-8m^2+48m+72m^2{m-1}^2\\ +96m{m-1}^3+36{m-1}^4+72m^{2}m-1+252m{m-1}^2\\ +144{m-1}^3+156mm-1+180{m-1}^2-72\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+62m+2b+32m+2b+3\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+612b+62m+2b+22m\\ +2b+2+18m4m+14m+2\text{,}\\ {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=340m^4+320m^3+2m^2+40m+72m^2{m-1}^2\\ +96m{m-1}^3+36{m-1}^4+72m^{2}m-1+252m{m-1}^2\\ +144{m-1}^3+156mm-1+180{m-1}^2-72\\ +{\displaystyle \sum }_{b=0}^{m-1}12b+612b+62m+2b\\ +2+18m4m+14m+2\text{,}\\ {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=408m^4+360m^3-6m^2+36m+72m^2{m-1}^2\\ +96m{m-1}^3+36{m-1}^4+72m^{2}m-1\\ +252m{m-1}^2+144{m-1}^3+156mm-1\\ +180{m-1}^2-72+18m4m+14m+2\text{,}\\ {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=408m^4+360m^3-6m^2+36m+72m^4-144m^3\\ +72m^2+96m^4-288m^3+288m^2-96m\\ +36m^4-144m^3+216m^2-144m+36\\ +72m^3-72m^2+252m^3-504m^2+252m\\ +144m^3-432m^2+432m-144+156m^2\\ -156m+180m^2-360m+180\\ -72+288m^3+216m^2+36m\text{,}\\ {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=408m^4+72m^4+96m^4+36m^4\\ +360m^3-144m^3-288m^3-144m^3+72m^3+252m^3\\ +144m^3+288m^3-6m^2+72m^2+288m^2+216m^2\\ -72m^2-504m^2-432m^2+156m^2+180m^2+216m^2\\ +36m-96m-144m+252m+432m-156m\\ -360m+36m+36-144+180-72\text{.}\end{array}$$

After calculation, we have$$\begin{array}{}\text{(24)}& {\varpi }_2{P}_r\mathrm{O}\mathrm{H}m=612m^4+540m^3+114m^2\text{.}\end{array}$$

(i) The comparison of ${\mathrm{G}\mathrm{A}}_4$ and ${\mathrm{A}\mathrm{B}\mathrm{C}}_5$ indices for prism octahedron network is shown in Figure 2 and Table 3.

[figure(s) omitted; refer to PDF]

Table 3

Comparison of ${\mathrm{G}\mathrm{A}}_4$ and ${\mathrm{A}\mathrm{B}\mathrm{C}}_5$ for prism octahedron network.

$\mathrm{D}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}m$	Range (b)	${\mathrm{A}\mathrm{B}\mathrm{C}}_5{\mathrm{P}}_{\mathrm{r}}\mathrm{O}\mathrm{H}m$	${\mathrm{G}\mathrm{A}}_4{\mathrm{P}}_{\mathrm{r}}\mathrm{O}\mathrm{H}m$
2	1	$163.0588$	$257.3122$
3	2	$454.32$	$714.4642$
4	3	$854.1818$	$1352.4273$
5	4	$1362.2926$	$2170.7695$
6	5	$1978.5306$	$3169.3163$

3. Conclusion

In this article, we computed the topological properties which are based on distance, of prism octahedron network of dimension $m$ such as total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to utilize the distance between the vertices of a prism octahedron network. Octahedron networks have a variety of useful applications in pharmacy, electronics, and networking. Individuals working in computer science and chemistry may find these findings beneficial from a chemical point of view. There are a number of unsolved problems in the evaluation of associated derived networks.

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