Analysis of Eccentricity-Based Topological Invariants with Zero-Divisor Graphs

Abstract

Let $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$ be a commutative ring, where ${♭}_1,{♭}_2,{♭}_3$ are distinct primes, and $q$ is any prime integer. A zero divisor graph $JR$ of ring $R$ is a graph with vertex set consist of zero divisors elements of $R$ and any two vertices $a,b$ are adjacent if and only if $ab=0$. A topological index is a numerical number associated with the graph and may be helpful to correlate the graph with certain of its physical/chemical properties. In this paper, we have computed some eccentricity based topological indices of $JR$, namely, atom-bond connectivity index ($\text{AB}{\text{C}}_5$), eccentricity-based harmonic index of fourth type ($H_{4}J$), geometric-arithmetic eccentricity index ($G{A}_{4}J$), eccentricity-based third Zagreb index, and eccentricity-based first Zagreb index.

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1. Introduction

There has been done wide investigation in combinatorics because of their strong ties to number theory and representation theory regarding algebraic structures [1, 2]. Due to their applications, finite rings and finite fields have attracted concentration in coding theory, cryptography along with vast theoretical study in these domains. The important functions known as molecular descriptors treat molecules as actual models as well as convert these molecules into numerals. These numerals are known as topological indices and are graph invariants.

Computing topological indices for different structures have been a study focus of recent work. In mathematical chemistry, the graph structure form of a chemical formula is called molecular graph. The compound’s atoms are considered as vertices, and chemical bonds between the vertices are considered as edges. A topological index can be defined as a numeric number that describes the topological structure of a chemical graph in a chemical graph while being unchanged under graph automorphism. As a result, there are several applications of these indices in nanotube structures, chemistry, and medical sciences [3, 4].

Topological indices are mainly characterized in to three categories: distance-based, degree-based, and counting-related topological indices [5–8]. Atom-bond connectivity index (ABC), Randic connectivity index ($R_{-1/2}$), geometric-arithmetic index (GA), and harmonic index (H) are some famous degree-based topological indices. The Estrada index, Wiener index, and Hosaya index are the topological indices which are based on distance [9, 10]. The eccentricity-based connectivity atom-bond index [11], eccentricity-based harmonic index of fourth type [12, 13], geometric-arithmetic eccentricity index [14], and Zagreb eccentricity index [15, 16] are few examples of eccentricity-based topological indices. Topological indices may be used to broaden interdisciplinary study through mathematical operations on graphs as well as define conditions under which chemical structures are formed.

These topological indices attached to a molecular graph are useful to predict certain of its physical and chemical properties. For instance, the ABC index is used to study the stability of branched and linear alkanes. This index is utilized to calculate the shear energy of cycloalkanes [17, 18]. The geometric-arithmetic index has strong ability of predict certain physical and chemical properties of chemical structure as compared to the Randic connectivity index [19, 20]. First and second Zagreb indices are were used to estimate the total $\pi$-electron energy of alternate hydrocarbons [21]. For investigation of the chemical properties of different molecular structures, degree-based topological indices have great importance. Such application of degree-based indices offers motivation to study eccentricity-based topological indices. Eccentricity-based topological indices can be used to assess a compound’s pharmacological, physicochemical, and toxicological qualities based on its molecular structure [22, 23]. To get more knowledge about topological indices and their applications, see [24–28].

2. Preliminary Definitions

Let $G=V,E$ be a graph, where $V$ denotes the vertex set, and $E$ denotes the edge set. Any two vertices $x,y\in V$ are adjacent if there is an edge between $x$ and $y$. The degree of a vertex $x$ is denoted by $d_x$ and is defined as the number of vertices adjacent to $x$. The distance between two vertices $x,y\in V$ is the length of the shortest path joining them. Eccentricity of a vertex $x$ is denoted by ${\epsilon }_x$ and is the maximum of the distances of all vertices from $x$. Mathematically ${\epsilon }_x=\max dx,y\text{:}y\in V$. To read more about the basic terminologies related to graph theory, see [29].

Let $R$ be a commutative ring with identity. Any two nonzero elements $x,y\in R$ are called zero divisors if $x.y=0$. Let $ZR$ be set of zero divisors of $R$. I. Beck [2] established the idea of the zero divisor graph $JR$ by considering $ZR$ as vertex set, and any two vertices $x,y\in ZR$ are connected by an edge if and only if $x.y=0$. His fundamental goal was to show how a commutative ring can be coloured [2]. Livingston and Anderson [30] proved that $JR$ is a connected graph. For more results on this topic, see [24, 31, 32].

In general, any eccentricity-based topological invariant, denoted by $TG$, is defined as $$\begin{array}{}\text{(1)}& TG=\sum _{xy\in EG}\varnothing {\epsilon }_x,{\epsilon }_y,\end{array}$$

where $\varnothing {\epsilon }_x,{\epsilon }_y$ is a real function with the property that $\varnothing {\epsilon }_x,{\epsilon }_y=\varnothing {\epsilon }_y,{\epsilon }_x$. From Equation (1), we can get some eccentricity based topological indices in the following way:

(a) Atom-bond connectivity eccentricity index $\text{AB}{\text{C}}_5$ if $\varnothing {\epsilon }_x,{\epsilon }_y=\sqrt{{\epsilon }_x+{\epsilon }_y-2/{\epsilon }_x\times {\epsilon }_y}$ [11]

(b) Eccentricity-based harmonic index of fourth type $H_4$ if $\varnothing {\epsilon }_x,{\epsilon }_y=2/{\epsilon }_x+{\epsilon }_y$ [12, 13]

(c) Geometric-arithmetic eccentricity index $G{A}_4$ if $\varnothing {\epsilon }_x,{\epsilon }_y=2\sqrt{{\epsilon }_x\times {\epsilon }_y}/{\epsilon }_x+{\epsilon }_y$ [14].

(d) First Zagreb eccentricity index $M_1^{\ast }$ if $\varnothing {\epsilon }_x,{\epsilon }_y={{\epsilon }_x+{\epsilon }_y}^{\beta }$, where $\beta =1$[15, 16]

(e) Third Zagreb eccentricity index $M_3^{\ast }$ if $\varnothing {\epsilon }_x,{\epsilon }_y={{\epsilon }_x\times {\epsilon }_y}^{\alpha }$, where $\alpha =1$ [15, 16]

3. Main Results

In this section, we compute the eccentricity based topological indices of a commutative ring $Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, where ${♭}_1,{♭}_2,{♭}_3$ are distinct primes, and $q$ is any prime integer.

Theorem 1.

Let ${♭}_1,{♭}_2,{♭}_3$ be distinct prime integers and $q$ be any prime number. Then for the zero divisor graph $JR$ of a ring $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, we have $V{J}_R={♭}_1{♭}_2{♭}_{3}q+{♭}_1{♭}_3{q}^2+{♭}_1{♭}_2{q}^2+{♭}_2{♭}_3{q}^2+{♭}_{1}q+2{♭}_2+{♭}_{3}q+q^2-{♭}_1{q}^2-{♭}_2{q}^2-{♭}_3{q}^2-{♭}_1{♭}_{3}q-{♭}_2{♭}_{3}q+{{♭}_1}^{2}q-3$.

Proof.

We can partition the vertex set of $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$ as follows:

(1) ${\varpi }_1=0,v\mid v\in Z_{q^2}:v\ne kq,0\le k\le q-1$. The vertices adjacent to $x\in {\varpi }_1$ are of the form $u^{\prime },0$ with $u^{\prime }\in Z_{{♭}_1{♭}_2{♭}_3}\backslash 0$. Hence, $d_x={♭}_1{♭}_2{♭}_3-1$ for all $x\in {\varpi }_1$ and ${\varpi }_1=q^2-q$

(2) ${\varpi }_2=0,v\mid v=kq,1\le k\le q-1$. So we have ${\varpi }_2=q-1$. The vertices adjacent to $x\in {\varpi }_2$ are of the form:

(i) $u^{\prime },0\text{:}{u}^{\prime }\in Z_{{♭}_1{♭}_2{♭}_3}\backslash 0$

(ii) $u^{\prime },v^{\prime }\text{:}{u}^{\prime }\in Z_{{♭}_1{♭}_2{♭}_3},v^{\prime }=t^{\prime }q,1\le t^{\prime }\le q-1$

Hence, for any $x\in {\varpi }_2$, $d_x={♭}_1{♭}_2{♭}_3-1+{♭}_1{♭}_2{♭}_3-1q-1={♭}_1{♭}_2{♭}_{3}q-q$.

(3) ${\varpi }_3=u,0\mid u=k{♭}_1{♭}_2;1\le k\le {♭}_3-1$ with ${\varpi }_3={♭}_3-1$. The vertices adjacent to $x\in {\varpi }_3$ are of the form:

(i) $0,v_1\text{:}{v}_1\in Z_{q^2}\backslash 0$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_1{♭}_3;1\le t_1\le {♭}_1{♭}_2-1$

(iii) $u^{\prime },v_2\text{:}{u}^{\prime }=t_1{♭}_3,v_2=t_{2}q;1\le t_1\le {♭}_1{♭}_2-1,1\le t_2\le q-1$

(iv) $u^{\prime },v_3\text{:}{u}^{\prime }=t_1{♭}_3,v_3\ne t_{3}q;1\le t_1\le {♭}_1{♭}_2-1,1\le t_3\le q-1$

Hence, for any $x\in {\varpi }_3$, $d_x=q^2-1+{♭}_1{♭}_2-1+{♭}_1{♭}_2-1q-1+{♭}_1{♭}_2-1q^2-q={♭}_1{♭}_2{q}^2-1$.

(4) ${\varpi }_4=u,0\text{:}u=k{♭}_1{♭}_3;1\le k\le {♭}_2-1.$ with ${\varpi }_4={♭}_2-1$. The vertices adjacent to $x\in {\varpi }_4$ are of the form:

(i) $0,v_1\text{:}{v}_1\in Z_{q^2}\backslash 0$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_1{♭}_2;1\le t_1\le {♭}_1{♭}_3-1$

(iii) $u^{\prime },v_2\text{:}{u}^{\prime }=t_1{♭}_2,v_2=t_{2}q;1\le t_1\le {♭}_1{♭}_3-1,1\le t_2\le q-1$

(iv) $u^{\prime },v_3\text{:}{u}^{\prime }=t_1{♭}_2,v_3\ne t_{3}q;1\le t_1\le {♭}_1{♭}_3-1,1\le t_3\le q-1$

Hence, for any $x\in {\varpi }_4$, $d_x=q^2-1+{♭}_1{♭}_3-1+{♭}_1{♭}_3-1q-1+{♭}_1{♭}_3-1q^2-q={♭}_1{♭}_3{q}^2-1$.

(5) ${\varpi }_5=u,0\text{:}u=k_1{♭}_1;1\le k_1\le {♭}_2{♭}_3-1;u\ne k_2{♭}_1{♭}_2,1\le k_2\le {♭}_3-1;u\ne k_3{♭}_1{♭}_3,1\le k_3\le {♭}_2-1$ with ${\varpi }_5={♭}_2-1{♭}_3-1$. The vertices adjacent to $x\in {\varpi }_5$ are of the form:

(i) $0,v_1\text{:}{v}_1\in Z_{q^2}\backslash 0$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_1{♭}_2{♭}_3;1\le t_1\le {♭}_1-1$

(iii) $u^{\prime },v_2\text{:}{u}^{\prime }=t_1{♭}_2{♭}_3,v_2=t_{2}q;1\le t_1\le {♭}_1-1,1\le t_2\le q-1$

(iv) $u^{\prime },v_3\text{:}{u}^{\prime }=t_1{♭}_2{♭}_3,v_3\ne t_{3}q;1\le t_1\le {♭}_1-1,1\le t_3\le q-1$

Hence, for any $x\in {\varpi }_5$, $d_x=q^2-1+{♭}_1-1+{♭}_1-1q-1+{♭}_1-1q^2-q={♭}_1{q}^2-1$.

(6) ${\varpi }_6=u,0;u=k{♭}_2{♭}_3,1\le k\le {♭}_1-1$ with ${\varpi }_6={♭}_1-1$. The vertices adjacent to $x\in {\varpi }_6$ are of the form:

(i) $0,v_1\text{:}{v}_1\in Z_{q^2}\backslash 0$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_1{♭}_1;1\le t_1\le {♭}_2{♭}_3-1$

(iii) $u^{\prime },v_2\text{:}{u}^{\prime }=t_1{♭}_1,v_2=t_{2}q;1\le t_1\le {♭}_2{♭}_3-1,1\le t_2\le q-1$

(iv) $u^{\prime },v_3\text{:}{u}^{\prime }=t_1{♭}_1,v_3\ne t_{3}q;1\le t_1\le {♭}_2{♭}_3-1,0\le t_3\le q-1$

Hence, for any $x\in {\varpi }_6$, $d_x=q^2-1+{♭}_2{♭}_3-1+{♭}_2{♭}_3-1q-1+{♭}_2{♭}_3-1q^2-q={♭}_2{♭}_3{q}^2-1$.

(7) ${\varpi }_7=u,0;u=k_1{♭}_2,1\le k_1\le {♭}_1{♭}_3-1;u\ne k_2{♭}_1{♭}_2,1\le k_2\le {♭}_3-1;u\ne k_3{♭}_2{♭}_3,1\le k_3\le {♭}_1-1$. So, ${\varpi }_7={♭}_1-1{♭}_3-1$. The vertices adjacent to $x\in {\varpi }_7$ are of the form:

(i) $0,v_1\text{:}{v}_1\in Z_{q^2}\backslash 0$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_1{♭}_1{♭}_3;1\le t_1\le {♭}_2-1$

(iii) $u^{\prime },v_2\text{:}{u}^{\prime }=t_1{♭}_1{♭}_3,v_2=t_{2}q;1\le t_1\le {♭}_2-1,1\le t_2\le q-1$

(iv) $u^{\prime },v_3\text{:}{u}^{\prime }=t_1{♭}_1{♭}_3,v_3\ne t_{3}q;1\le t_1\le {♭}_2-1,0\le t_3\le q-1$

Hence, for any $x\in {\varpi }_7$, $d_x=q^2-1+{♭}_2-1+{♭}_2-1q-1+{♭}_2-1q^2-q={♭}_2{q}^2-1$.

(8) ${\varpi }_8=u,0;u=k_1{♭}_3,1\le k_1\le {♭}_1{♭}_2-1;u\ne k_2{♭}_1{♭}_{3}1\le k_2\le {♭}_2-1;u\ne k_3{♭}_2{♭}_3,1\le k_3\le {♭}_1-1,$ with ${\varpi }_8={♭}_1-1{♭}_2-1$. The vertices adjacent to $x\in {\varpi }_8$ are of the form:

(i) $0,v_1\text{:}{v}_1\in Z_{q^2}\backslash 0$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_1{♭}_1{♭}_2;1\le t_1\le {♭}_3-1$

(iii) $u^{\prime },v_2\text{:}{u}^{\prime }=t_1{♭}_1{♭}_2,v_2=t_{2}q;1\le t_1\le {♭}_3-1,1\le t_2\le q-1$

(iv) $u^{\prime },v_3\text{:}{u}^{\prime }=t_1{♭}_1{♭}_2,v_3\ne t_{3}q;1\le t_1\le {♭}_3-1,0\le t_3\le q-1$

Hence, for any $x\in {\varpi }_7$, $d_x=q^2-1+{♭}_3-1+{♭}_3-1q-1+{♭}_3-1q^2-q={♭}_3{q}^2-1$.

(9) ${\varpi }_9=u,0;u\ne k_1{♭}_1,1\le k_1\le {♭}_2{♭}_3-1;u\ne k_2{♭}_2,1\le k_2\le {♭}_1{♭}_3-1;u\ne k_3{♭}_3,1\le k_3\le {♭}_1{♭}_2-1$ with ${\varpi }_9={♭}_1-1{♭}_2-1{♭}_3-1$. In this case $d_x=q^2-1$ for all $x\in {\varpi }_9$

(10) ${\varpi }_{10}=u,v\mid u=k_1{♭}_1{♭}_2,1\le k_1\le {♭}_3-1;v\ne k_{2}q,0\le k_2\le q-1$ with ${\varpi }_{10}={♭}_3-1q^2-q$. In this case $d_x={♭}_1{♭}_2-1$ for all $x\in {\varpi }_{10}$

(11) ${\varpi }_{11}=u,v\mid u=k_1{♭}_1{♭}_3,1\le k_1\le {♭}_2-1;v\ne k_{2}q,0\le k_2\le q-1,$ with ${\varpi }_{11}={♭}_2-1q^2-q$. In this case $d_x={♭}_1{♭}_3-1$ for all $x\in {\varpi }_{11}$

(12) ${\varpi }_{12}=u,v\mid u=k_1{♭}_1,1\le k_1\le {♭}_2{♭}_3-1;u\ne k_2{♭}_1{♭}_2,1\le k_2\le {♭}_3-1;u\ne k_3{♭}_1{♭}_3,1\le k_3\le {♭}_2-1;v\ne k_{4}q,0\le k_4\le q-1$ with ${\varpi }_{12}={♭}_2-1{♭}_3-1q^2-q$. The vertices adjacent to $x\in {\varpi }_{12}$ are of the form $u^{\prime },0;u^{\prime }=t^{\prime }{♭}_2{♭}_3,1\le t^{\prime }\le {♭}_1-1$. Hence $d_x={♭}_1-1$ for all $x\in {\varpi }_{12}$

(13) ${\varpi }_{13}=u,v\mid u=k_1{♭}_2{♭}_3,1\le k_1\le {♭}_1-1;v\ne k_{2}q,0\le k_2\le q-1$ with ${\varpi }_{13}={♭}_1-1q^2-q$. The vertices adjacent to $x\in {\varpi }_{12}$ are of the form $u^{\prime },0;u^{\prime }=t^{\prime }{♭}_1,1\le t^{\prime }\le {♭}_2{♭}_3-1$. Hence $d_x={♭}_2{♭}_3-1$ for all $x\in {\varpi }_{13}$

(14) ${\varpi }_{14}=u,v;u=k_1{♭}_2,1\le k_1\le {♭}_1{♭}_3-1;u\ne k_2{♭}_1{♭}_2,1\le k_2\le {♭}_3-1;u\ne k_3{♭}_2{♭}_3,1\le k_3\le {♭}_1-1;v\ne k_{4}q,0\le k_4\le q-1$ with ${\varpi }_{14}={♭}_1-1{♭}_3-1q^2-q$. Then, each vertex $x=\in {\varpi }_{14}$ has degree ${♭}_2-1$

(15) ${\varpi }_{15}=u,v\mid u=k_1{♭}_3,1\le k_1\le {♭}_1{♭}_2-1;u\ne k_2{♭}_1{♭}_{3}1\le k_2\le {♭}_2-1;u\ne k_3{♭}_2{♭}_3,1\le k_3\le {♭}_1-1;v\ne k_{4}q,0\le k_4\le q-1,$ with ${\varpi }_{15}={♭}_1-1{♭}_2-1q^2-q$. The vertices adjacent to $x\in {\varpi }_{15}$ are of the form $u^{\prime },0;u^{\prime }=t^{\prime }{♭}_1{♭}_2,1\le t^{\prime }\le {♭}_3-1$. Then each vertex $x=\in {\varpi }_{15}$ has degree ${♭}_3-1$

(16) ${\varpi }_{16}=u,v\mid u=k_1{♭}_1{♭}_2;1\le k_1\le {♭}_3-1;v=k_{2}q,1\le k_2\le q-1$ with ${\varpi }_{16}={♭}_3-1q-1$. The vertices adjacent to $x\in {\varpi }_{16}$ are of the form:

(i) $0,v_1\text{:}{v}_1=t_{1}q,1\le t_1\le q-1$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_2{♭}_3;1\le t_2\le {♭}_1{♭}_2-1$

(iii) $u^{\prime },v_1\text{:}{u}^{\prime }=t_2{♭}_3,v_1=t_{1}q;1\le t_1\le q-1,1\le t_2\le {♭}_1{♭}_2-1$

Hence, for any $x\in {\varpi }_{16}$, $d_x=q-1+{♭}_1{♭}_2-1+{♭}_1{♭}_2-1q-1={♭}_1{♭}_{2}q-1$.

(17) ${\varpi }_{17}=u,v\mid u=k_1{♭}_1{♭}_3,1\le k_1\le {♭}_2-1;v=k_{2}q,1\le k_2\le q-1$ with ${\varpi }_{17}={♭}_2-1$. The vertices adjacent to $x\in {\varpi }_{17}$ are of the form:

(i) $0,v_1\text{:}{v}_1=t_{1}q,1\le t_1\le q-1$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_2{♭}_2,1\le t_2\le {♭}_1{♭}_3-1$

(iii) $u^{\prime },v_1\text{:}{v}_1=t_{1}q,u^{\prime }=t_2{♭}_2:1\le t_1\le q-1,1\le t_2\le {♭}_1{♭}_3-1$

Hence, for any $x\in {\varpi }_{17}$, $d_x=q-1+{♭}_1{♭}_3-1+{♭}_1{♭}_3-1q-1={♭}_1{♭}_{3}q-1$.

(18) ${\varpi }_{18}=u,v\mid u=k_1{♭}_1,1\le k_1\le {♭}_2{♭}_3-1;u\ne k_2{♭}_1{♭}_2,1\le k_2\le {♭}_3-1;u\ne k_3{♭}_1{♭}_3,1\le k_3\le {♭}_2-1;v=k_{4}q,1\le k_4\le q-1$ with ${\varpi }_{18}={♭}_2-1{♭}_3-1q-1$. The vertices adjacent to $x\in {\varpi }_{18}$ are of the form:

(i) $u^{\prime },v^{\prime }\text{:}{u}^{\prime }=t_1{♭}_2{♭}_3,1\le t_1\le {♭}_1-1;v^{\prime }=t_{2}q,0\le t_2\le q-1$

(ii) $0,v_1\text{:}{v}_1=t_{3}q,1\le t_3\le q-1$

Hence, for any $x\in {\varpi }_{18}$, $d_x={♭}_1-1q^2-q+q-1=q-1{♭}_{1}q-q+1$.

(19) ${\varpi }_{19}=u,v\mid u=k_1{♭}_2{♭}_3,1\le k_1\le {♭}_1-1;v=k_{2}q,1\le k_2\le q-1$ with ${\varpi }_{19}={♭}_1-1q-1$. The vertices adjacent to $x\in {\varpi }_{19}$ are of the form:

(i) $0,v_1\text{:}{v}_1=t_{1}q,1\le t_1\le q-1$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_2{♭}_1,1\le t_2\le {♭}_2{♭}_3-1$

(iii) $u^{\prime },v_1\text{:}{u}^{\prime }=t_2{♭}_1,v_1=t_{1}q;1\le t_1\le q-1,1\le t_2\le {♭}_2{♭}_3-1$

Hence, for any $x\in {\varpi }_{19}$, $d_x=q-1+{♭}_2{♭}_3-1+{♭}_2{♭}_3-1q-1={♭}_2{♭}_{3}q-1$.

(20) ${\varpi }_{20}=u,v\mid u=k_1{♭}_2,1\le k_1\le {♭}_1{♭}_3-1;u\ne k_2{♭}_1{♭}_2,1\le k_2\le {♭}_3-1;u\ne k_3{♭}_2{♭}_3,1\le k_3\le {♭}_1-1;v=k_{4}q,1\le k_4\le q-1$ with ${\varpi }_{20}={♭}_1-1{♭}_3-1q-1$. The vertices adjacent to $x\in {\varpi }_{20}$ are of the form:

(i) $0,v_1\text{:}{v}_1=t_{1}q,1\le t_1\le q-1$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_2{♭}_1{♭}_3;1\le t_2\le {♭}_2-1$

(iii) $u^{\prime },v_1\text{:}{u}^{\prime }=t_2{♭}_1{♭}_3,v_1=t_{1}q;1\le t_1\le q-1,1\le t_2\le {♭}_2-1$

Hence, for any $x\in {\varpi }_{20}$, $d_x=q-1+{♭}_2-1+{♭}_2-1q-1={♭}_{2}q-1$.

(21) ${\varpi }_{21}=u,v;u=k_1{♭}_3,1\le k_1\le {♭}_1{♭}_2-1;u\ne k_2{♭}_1{♭}_{3}1\le k_2\le {♭}_2-1;u\ne k_3{♭}_2{♭}_3,1\le k_3\le {♭}_{1}1;v=k_{4}q,1\le k_4\le q-1$ with ${\varpi }_{21}={♭}_1-1{♭}_2-1q-1$. The vertices adjacent to $x\in {\varpi }_{21}$ are of the form:

(i) $0,v_1\text{:}{v}_1=t_{1}q,1\le t_1\le q-1$

(ii) $u^{\prime },0\text{:}{u}^{\prime }=t_2{♭}_1{♭}_2,1\le t_2\le {♭}_3-1$

(iii) $u^{\prime },v_1\text{:}{u}^{\prime }=t_2{♭}_1{♭}_2,v_1=t_{1}q;1\le t_1\le q-1,1\le t_2\le {♭}_3-1$

Hence, for any $x\in {\varpi }_{21}$, $d_x=q-1+{♭}_3-1+{♭}_3-1q-1={♭}_{3}q-1$.

(22) ${\varpi }_{22}=u,v\mid u\ne k_1{♭}_1,1\le k_1\le {♭}_2{♭}_3-1;u\ne k_2{♭}_2,1\le k_2\le {♭}_1{♭}_3-1;u\ne k_3{♭}_3,1\le k_3\le {♭}_1{♭}_2-1;v=k_{4}q,1\le k_4\le q-1$. with ${\varpi }_{22}={♭}_1-1{♭}_2-1{♭}_3-1q-1$. Then each vertex $x=\in {\varpi }_{22}$ has degree $q-1$

Now, the cardinality of vertex set $V{J}_R$ can be calculated as $V{J}_R=q^2-q+q-1+{♭}_3-1+{♭}_2-1+{♭}_2-1{♭}_3-1+{♭}_1-1+{♭}_1-1{♭}_3-1+{♭}_1-1{♭}_2-1+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_3-1q^2-q+{♭}_2-1q^2-q+{♭}_2-1{♭}_3-1q^2-q+{♭}_1-1q^2-q+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+{♭}_3-1q-1+{♭}_2-1+{♭}_2-1{♭}_3-1q-1+{♭}_1-1q-1+{♭}_1-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1={♭}_1{♭}_2{♭}_{3}q+{♭}_1{♭}_3{q}^2+{♭}_1{♭}_2{q}^2+{♭}_2{♭}_3{q}^2+{♭}_{1}q+2{♭}_2+{♭}_{3}q+q^2-{♭}_1{q}^2-{♭}_2{q}^2-{♭}_3{q}^2-{♭}_1{♭}_{3}q-{♭}_2{♭}_{3}q+{{♭}_1}^{2}q-3$.

From the above Theorem, we can compute the number of edges $E{J}_R$ by using hand shaking lemma. $$\begin{array}{}\text{(2)}& E{J}_R=q^2-q{♭}_1{♭}_2{♭}_3-1+q-1{♭}_1{♭}_2{♭}_{3}q-q+{♭}_3-1{♭}_1{♭}_2{q}^2-1+{♭}_2-1{♭}_1{♭}_3{q}^2-1+{♭}_2-1{♭}_3-1{♭}_1{q}^2-1+{♭}_1-1{♭}_2{♭}_3{q}^2-1+{♭}_1-1{♭}_3-1{♭}_2{q}^2-1+{♭}_1-1{♭}_2-1{♭}_3{q}^2-1+{♭}_1-1{♭}_2-1{♭}_3-1q^2-1+{♭}_3-1q^2-q{♭}_1{♭}_2-1+{♭}_2-1q^2-q{♭}_1{♭}_3-1+{♭}_2-1{♭}_3-1q^2-q{♭}_1-1+{♭}_1-1q^2-q{♭}_2{♭}_3-1+{♭}_1-1{♭}_3-1q^2-q{♭}_2-1+{♭}_1-1{♭}_2-1q^2-q{♭}_3-1+{♭}_3-1q-1{♭}_1{♭}_{2}q-1+{♭}_2-1{♭}_1{♭}_{3}q-1+{♭}_2-1{♭}_3-1q-1q-1{♭}_{1}q-q+1+{♭}_1-1q-1{♭}_2{♭}_{3}q-1+{♭}_1-1{♭}_3-1q-1{♭}_{2}q-1+{♭}_1-1{♭}_2-1q-1{♭}_{3}q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1q-1.=2-q-1+q+q^2+-2+q-1+q3+q{♭}_2+q-2+q3+q--6+q9+q{♭}_2{♭}_3+q{♭}_1-2+q3+q--4+q8+q{♭}_3+{♭}_{2}6-q9+q+-12+q18+q{♭}_2.\end{array}$$

Anderson and Livingston [30] proved that the diameter of the graph $JR$ is atmost three. It follows that the eccentricity of any vertex $x\in VJR$ is atmost three. Figure 1 reflects this fact for the case $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$.

Theorem 2 (see [30]).

Let $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, where ${♭}_1,{♭}_2,{♭}_3$ be distinct prime integers and $q$ is any prime number. Then, eccentricity of a vertex $x\in VJR$ is either 2 or 3.

In the next Theorem, we find the exact expressions for the $\text{AB}{\text{C}}_5$, $H_4$, $G{A}_4$, $M_1^{\ast }$$M_3^{\ast }$ indices of $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, where ${♭}_1,{♭}_2,{♭}_3$ be distinct prime integers and $q$ is any prime number. For simplicity, we fix some notations. Let $\alpha =6-3q+-3+q{♭}_3+{♭}_2-3+q+{♭}_3+{♭}_1-3+q+{♭}_2+{♭}_3$, $$\begin{array}{}\text{(3)}& \beta =-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3,\text{(4)}& \gamma ={♭}_{3}4+q-1+3{-1+q}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3,\text{(5)}& \delta ={♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3,\text{(6)}& \zeta =-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3,\text{(7)}& \mu ={♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3.\end{array}$$

Theorem 3.

Let $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, where ${♭}_1,{♭}_2,{♭}_3$ be distinct prime integers, and $q$ is any prime number. Then, $TG$ has the following expression $T{J}_R=\alpha \varphi 2,2+\beta +\gamma +\delta \varphi 2,3+\zeta +\mu \varphi 3,3$,

Proof.

Let $$\begin{array}{}\text{(8)}& {\Xi }_{\text{m},\text{n}}=u^{\prime }{v}^{\prime }\in E{J}_R\text{:}{\epsilon }_{u^{\prime }}=m,{\epsilon }_{v^{\prime }}=n\end{array}$$

Then, from Theorem 1 and Figure 1, we have $$\begin{array}{}\text{(9)}& {\Xi }_{2,2}=q-1{♭}_3-1+q-1{♭}_2-1+q-1p-1+{♭}_3-1{♭}_2-1+{♭}_3-1{♭}_1-1+{♭}_2-1{♭}_1-1,\text{(10)}& {\Xi }_{2,3}=q-1{♭}_2-1{♭}_3-1+q-1{♭}_1-1{♭}_3-1+q-1{♭}_1-1{♭}_2-1+q-1{♭}_1-1{♭}_2-1{♭}_3-1+{q-1}^2{♭}_3-1+q-1{♭}_2-1+{q-1}^2{♭}_2-1{♭}_3-1+{q-1}^2{♭}_1-1+{q-1}^2{♭}_1-1{♭}_3-1+{q-1}^2{♭}_1-1{♭}_2-1+{q-1}^2{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1+q^2-q{♭}_2-1{♭}_3-1+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+{♭}_2-1{♭}_3-1+{♭}_1-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_2-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_2-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_1-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+{♭}_1-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1\end{array}$$

and $$\begin{array}{}\text{(11)}& {\Xi }_{3,3}={♭}_2-1{♭}_3-1q^2-q+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+4{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+3{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_3-1{q-1}^2+2{♭}_1-1{♭}_2-1{♭}_3-1{q-1}^2+{♭}_1-1{♭}_2-1q-1.\end{array}$$

Finally, $$\begin{array}{c}\text{(12)}& \text{T}{\text{J}}_{\text{R}}=\sum _{u^{\prime }{v}^{\prime }\in E{J}_R}\varnothing {\epsilon }_{u^{\prime }},{\epsilon }_{v^{\prime }}=\sum _{u^{\prime }{v}^{\prime }\in {\Xi }_{2,2}}\varnothing 2,2+\sum _{u^{\prime }{v}^{\prime }\in {\Xi }_{2,3}}\varnothing 2,3+\sum _{u^{\prime }{v}^{\prime }\in {\Xi }_{3,3}}\varnothing 3,3=q-1{♭}_3-1+q-1{♭}_2-1+q-1p-1+{♭}_3-1{♭}_2-1+{♭}_3-1{♭}_1-1+{♭}_2-1{♭}_1-1\varnothing 2,2+q-1{♭}_2-1{♭}_3-1+q-1{♭}_1-1{♭}_3-1+q-1{♭}_1-1{♭}_2-1+q-1{♭}_1-1{♭}_2-1{♭}_3-1+{q-1}^2{♭}_3-1+q-1{♭}_2-1+{q-1}^2{♭}_2-1{♭}_3-1+{q-1}^2{♭}_1-1+{q-1}^2{♭}_1-1{♭}_3-1+{q-1}^2{♭}_1-1{♭}_2-1+{q-1}^2{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1+q^2-q{♭}_2-1{♭}_3-1+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+{♭}_2-1{♭}_3-1+{♭}_1-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_2-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_2-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_1-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+{♭}_1-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1+{♭}_1-1{♭}_2-1{♭}_3-1q-1\varnothing 2,3+{♭}_2-1{♭}_3-1q^2-q+{♭}_1-1{♭}_3-1q^2-q+{♭}_1-1{♭}_2-1q^2-q+4{♭}_1-1{♭}_2-1{♭}_3-1q^2-q+3{♭}_1-1{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_2-1{♭}_3-1+{♭}_2-1{♭}_3-1q-1+{♭}_1-1{♭}_3-1{q-1}^2+2{♭}_1-1{♭}_2-1{♭}_3-1{q-1}^2+{♭}_1-1{♭}_2-1q-1\varnothing 3,3.=6-3q+-3+q{♭}_3+{♭}_2-3+q+{♭}_3+{♭}_1-3+q+{♭}_2+{♭}_3\varphi 2,2+-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3+{♭}_{3}4+q-1+3{-1+q}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3\varphi 2,3+-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3+{♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3\varphi 3,3,=\alpha \varphi 2,2+\beta +\gamma +\delta \varphi 2,3+\zeta +\mu \varphi 3,3.\end{array}$$

Theorem 4.

Let $R=Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, where ${♭}_1,{♭}_2,{♭}_3$ be distinct prime integers and $q$ is any prime number. Then $$\begin{array}{}\text{(13)}& M_1^{\ast }{J}_R=4\alpha +\beta +\gamma +\delta 5+\zeta +\mu 6,\text{(14)}& M_3^{\ast }{J}_R=4\alpha +\beta +\gamma +\delta 6+\zeta +\mu 9,\text{(15)}& G{A}_4{J}_R=\alpha +\beta +\gamma +\delta \frac{2\sqrt{6}}{5}+\zeta +\mu ,\text{(16)}& \text{AB}{\text{C}}_5{J}_R=\frac{\alpha }{\sqrt{2}}+\beta +\gamma +\delta \frac{1}{\sqrt{2}}+\zeta +\mu \frac{2}{3},\text{(17)}& H_4{J}_R=\frac{\alpha }{2}+\beta +\gamma +\delta \frac{2}{5}+\zeta +\mu \frac{1}{3}.\end{array}$$

Proof.

For the first Zagreb eccentricity index, we have $\varnothing {\epsilon }_{u^{\prime }},{\epsilon }_{v^{\prime }}={\epsilon }_{u\prime }+{\epsilon }_{v\prime }$. Then, $\varnothing 2,2=4$, $\varnothing 2,3=5$, and $\varnothing 3,3=6$. Substituting these values in Theorem 3, we get $$\begin{array}{c}\text{(18)}& M_1^{\ast }{J}_R=6-3\text{q}+-3+\text{q}{♭}_3+{♭}_2-3+\text{q}+{♭}_3+{♭}_1-3+\text{q}+{♭}_2+{♭}_3\varphi 2,2+-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3+{♭}_{3}4+q-1+3-1+q{-1+{♭}_3}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3\varphi 2,3\\ +-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3+{♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3\varphi 3,3.\\ =6+{♭}_3-3+q-3q+{♭}_2-3+{♭}_3+q+{♭}_1-3+{♭}_2+{♭}_3+q4\\ +-4+{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2+{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2--2+qq1+q^2+{♭}_{3}4+-3+{♭}_3{♭}_3{-1+q}^2{q}^2+q-1+3{-1+q}^{2}q+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2-3+{♭}_3{♭}_3{-1+q}^2{q}^2-2q1+3{-1+q}^{2}q+{♭}_{1}3-2{♭}_3+q-1+2{-1+q}^{2}q+q-2{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^{2}q+{♭}_{3}q-7-2-3+{♭}_3{♭}_3{-1+q}^2-6-2+qq+{♭}_2-2-4{-1+q}^{2}q+{♭}_{3}q4-3+{♭}_3{♭}_3{-1+q}^2+35+4-2+qq5\\ +-1-2-1+qq+{♭}_{3}q-2+3q-{♭}_2-2+5-4qq+{♭}_{3}1+5-1+qq+{♭}_1{♭}_3+q-2+3q+{♭}_2-1-5-1+qq+{♭}_{3}q2-4q+{♭}_2-5+6q6,=4\alpha +\beta +\gamma +\delta 5+\zeta +\mu 6.\end{array}$$

[figure(s) omitted; refer to PDF]

For third Zagreb eccentricity index, we have $\varnothing {\epsilon }_{u\prime },{\epsilon }_{v\prime }={\epsilon }_{u\prime }\times {\epsilon }_{v\prime }$ and $$\begin{array}{c}\text{(19)}& M_3^{\ast }=6-3q+-3+q{♭}_3+{♭}_2-3+q+{♭}_3+{♭}_1-3+q+{♭}_2+{♭}_3\varphi 2,2+-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3+{♭}_{3}4+q-1+3{-1+q}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3\varphi 2,3+-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3+{♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3\varphi 3,3.=6+{♭}_3-3+q-3q+{♭}_2-3+{♭}_3+q+{♭}_1-3+{♭}_2+{♭}_3+q4+-4+{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2+{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2--2+qq1+q^2+{♭}_{3}4+-3+{♭}_3{♭}_3{-1+q}^2{q}^2+q-1+3{-1+q}^{2}q+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2-3+{♭}_3{♭}_3{-1+q}^2{q}^2-2q1+3{-1+q}^{2}q+{♭}_{1}3-2{♭}_3+q-1+2{-1+q}^{2}q+q-2{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^{2}q+{♭}_{3}q-7-2-3+{♭}_3{♭}_3{-1+q}^2-6-2+qq+{♭}_2-2-4{-1+q}^{2}q+{♭}_{3}q4-3+{♭}_3{♭}_3{-1+q}^2+35+4-2+qq6+-1-2-1+qq+{♭}_{3}q-2+3q-{♭}_2-2+5-4qq+{♭}_{3}1+5-1+qq+{♭}_1{♭}_3+q-2+3q+{♭}_2-1-5-1+qq+{♭}_{3}q2-4q+{♭}_2-5+6q9,=4\alpha +\beta +\gamma +\delta 6+\zeta +\mu 9.\end{array}$$

For the geometric-arithmetic eccentricity index, we have $\varnothing {\epsilon }_{u\prime },{\epsilon }_{v\prime }=2\sqrt{{\epsilon }_{u\prime }\times {\epsilon }_{v\prime }}/{\epsilon }_{u\prime }+{\epsilon }_{v\prime }$ and $$\begin{array}{c}\text{(20)}& G{A}_4{J}_R=6-3q+-3+q{♭}_3+{♭}_2-3+q+{♭}_3+{♭}_1-3+q+{♭}_2+{♭}_3\varphi 2,2+-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3+{♭}_{3}4+q-1+3{-1+q}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3\varphi 2,3+-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3+{♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3\varphi 3,3.=5+{♭}_3-3+q-3q-2-1+qq+{♭}_2-3+{♭}_3+q+{♭}_1-3+{♭}_2+{♭}_3+q+-4+{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2+{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2--2+qq1+q^2+{♭}_{3}4+-3+{♭}_3{♭}_3{-1+q}^2{q}^2+q-1+3{-1+q}^{2}q+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2-3+{♭}_3{♭}_3{-1+q}^2{q}^2-2q1+3{-1+q}^{2}q+{♭}_{1}3-2{♭}_3+q-1+2{-1+q}^{2}q+q-2{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^{2}q+{♭}_{3}q-7-2-3+{♭}_3{♭}_3{-1+q}^2-6-2+qq+{♭}_2-2-4{-1+q}^{2}q+{♭}_{3}q4-3+{♭}_3{♭}_3{-1+q}^2+35+4-2+qq\frac{2\sqrt{6}}{5}+{♭}_{3}q-2+3q-{♭}_2-2+5-4qq+{♭}_{3}1+5-1+qq+{♭}_1{♭}_3+q-2+3q+{♭}_2-1-5-1+qq+{♭}_{3}q2-4q+{♭}_2-5+6q,=\alpha +\beta +\gamma +\delta \frac{2\sqrt{6}}{5}+\zeta +\mu .\end{array}$$

For atom-bond connectivity eccentricity index, we have $\varnothing {\epsilon }_{u\prime },{\epsilon }_{v\prime }=\sqrt{{\epsilon }_{u\prime }+{\epsilon }_{v\prime }-2/{\epsilon }_{u\prime }\times {\epsilon }_{v\prime }}$ and $$\begin{array}{c}\text{(21)}& \text{AB}{\text{C}}_5{J}_R=6-3q+-3+q{♭}_3+{♭}_2-3+q+{♭}_3+{♭}_1-3+q+{♭}_2+{♭}_3\varphi 2,2+-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3+{♭}_{3}4+q-1+3{-1+q}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3\varphi 2,3+-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3+{♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3\varphi 3,3.=\frac{6+{♭}_3-3+q-3q+{♭}_2-3+{♭♭}_3+q+{♭}_1-3+{♭}_2+{♭}_3+q}{\sqrt{2}}+-4+{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2+{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2--2+qq1+q^2+{♭}_{3}4+-3+{♭}_3{♭}_3{-1+q}^2{q}^2+q-1+3{-1+q}^{2}q+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2-3+{♭}_3{♭}_3{-1+q}^2{q}^2-2q1+3{-1+q}^{2}q+{♭}_{1}3-2{♭}_3+q-1+2{-1+q}^{2}q+q-2{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^{2}q+{♭}_{3}q-7-2-3+{♭}_3{♭}_3{-1+q}^2-6-2+qq+{♭}_2-2-4{-1+q}^{2}q+{♭}_{3}q4-3+{♭}_3{♭}_3{-1+q}^2+35+4-2+qq\frac{1}{\sqrt{2}}+-1-2-1+qq+{♭}_{3}q-2+3q-{♭}_2-2+5-4qq+{♭}_{3}1+5-1+qq+{♭}_1{♭}_3+q-2+3q+{♭}_2-1-5-1+qq+{♭}_{3}q2-4q+{♭}_2-5+6q\frac{2}{3},=\frac{\alpha }{\sqrt{2}}+\beta +\gamma +\delta \frac{1}{\sqrt{2}}+\zeta +\mu \frac{2}{3}.\end{array}$$

For the harmonic index based on eccentricity of fourth type, we have $\varnothing {\epsilon }_{u\prime },{\epsilon }_{v\prime }=2/{\epsilon }_{u\prime }+{\epsilon }_{v\prime }$ and $$\begin{array}{c}\text{(22)}& H_4{J}_R=6-3q+-3+q{♭}_3+{♭}_2-3+q+{♭}_3+{♭}_1-3+q+{♭}_2+{♭}_3\varphi 2,2+-4--2+qq1+q^2+{-1+q}^2{q}^2{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3+{-1+q}^2{q}^2{{♭}_2}^2{-1+{♭}_3}^3+{♭}_{3}4+q-1+3{-1+q}^{2}q+{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2q1+3{-1+q}^{2}q-2{-1+q}^2{q}^2-3+{♭}_3{♭}_3+{♭}_{1}3+q-1+2{-1+q}^{2}q-2{♭}_3+q-2{-1+q}^{2}q{{♭}_2}^2{-1+{♭}_3}^3+q{♭}_3-7-6-2+qq-2{-1+q}^2-3+{♭}_3{♭}_3+{♭}_2-2-4{-1+q}^{2}q+q{♭}_{3}35+4-2+qq+4{-1+q}^2-3+{♭}_3{♭}_3\varphi 2,3+-1-2-1+qq+q-2+3q{♭}_3-{♭}_2-2+5-4qq+1+5-1+qq{♭}_3+{♭}_{1}q-2+3q+-1-5-1+qq{♭}_2+{♭}_3+q2-4q+-5+6q{♭}_2{♭}_3\varphi 3,3.=\frac{6+{♭}_3-3+q-3q+{♭}_2-3+{♭}_3+q+{♭}_1-3+{♭}_2+{♭}_3+q}{2}+-4+{{♭}_1}^2{-1+{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2+{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^2{q}^2--2+qq1+q^2+{♭}_{3}4+-3+{♭}_3{♭}_3{-1+q}^2{q}^2+q-1+3{-1+q}^{2}q+{♭}_{2}2q2+-2+q{q}^2+{♭}_3-1-2-3+{♭}_3{♭}_3{-1+q}^2{q}^2-2q1+3{-1+q}^{2}q+{♭}_{1}3-2{♭}_3+q-1+2{-1+q}^{2}q+q-2{{♭}_2}^2{-1+{♭}_3}^3{-1+q}^{2}q+{♭}_{3}q-7-2-3+{♭}_3{♭}_3{-1+q}^2-6-2+qq+{♭}_2-2-4{-1+q}^{2}q+{♭}_{3}q4-3+{♭}_3{♭}_3{-1+q}^2+35+4-2+qq\frac{2}{5}+-1-2-1+qq+{♭}_{3}q-2+3q-{♭}_2-2+5-4qq+{♭}_{3}1+5-1+qq+{♭}_1{♭}_3+q-2+3q+{♭}_2-1-5-1+qq+{♭}_{3}q2-4q+{♭}_2-5+6q\frac{1}{3},=\frac{\alpha }{2}+\beta +\gamma +\delta \frac{2}{5}+\zeta +\mu \frac{1}{3}.\end{array}$$

4. Conclusion

For zero divisor graph of commutative ring $Z_{{♭}_1{♭}_2{♭}_3}\times Z_{q^2}$, we calculated the eccentricity-based atom-bond index of connectivity, the harmonic index based on eccentricity of fourth type, the geometric-arithmetic eccentricity index, the eccentricity-based third Zagreb index, and the eccentricity-based first Zagreb index. This work is a part of the open problem to calculate the eccentricity based topological indices for zero divisor graph of commutative ring $Z_m\times Z_n$ for $m,n\in \mathbb{Z}$. In future, similar work can be done for other cases of commutative ring $Z_m\times Z_n$.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (grant no. 61702291) and China Henan International Joint Laboratory for Multidimensional Topology and Carcinogenic Characteristics Analysis of Atmospheric Particulate Matter PM2.5.

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