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

Authors: Zhi-hao Hui; Abdul Rauf; Muhammad Mohsin Abbas; Adnan Aslam
Journal: Journal of Function Spaces
 (2022)

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## Abstract

LetR=Z♭1♭2♭3×Zq2 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 (ABC5), eccentricity-based harmonic index of fourth type (H4J), geometric-arithmetic eccentricity index (GA4J), eccentricity-based third Zagreb index, and eccentricity-based first Zagreb index.


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## Body

## 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 π-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

LetG=V,E be a graph, where V denotes the vertex set, and E denotes the edge set. Any two vertices x,y∈V are adjacent if there is an edge between x and y. The degree of a vertex x is denoted by dx and is defined as the number of vertices adjacent to x. The distance between two vertices x,y∈V is the length of the shortest path joining them. Eccentricity of a vertex x is denoted by εx and is the maximum of the distances of all vertices from x. Mathematically εx=maxdx,y:y∈V. To read more about the basic terminologies related to graph theory, see [29].LetR be a commutative ring with identity. Any two nonzero elements x,y∈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∈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 byTG, is defined as
(1)TG=∑xy∈EG∅εx,εy,where∅εx,εy is a real function with the property that ∅εx,εy=∅εy,εx. From Equation (1), we can get some eccentricity based topological indices in the following way:
(a)
Atom-bond connectivity eccentricity indexABC5 if ∅εx,εy=εx+εy−2/εx×εy [11](b)
Eccentricity-based harmonic index of fourth typeH4 if ∅εx,εy=2/εx+εy [12, 13](c)
Geometric-arithmetic eccentricity indexGA4 if ∅εx,εy=2εx×εy/εx+εy [14].(d)
First Zagreb eccentricity indexM1∗ if ∅εx,εy=εx+εyβ, where β=1[15, 16](e)
Third Zagreb eccentricity indexM3∗ if ∅εx,εy=εx×εyα, where α=1 [15, 16]


## 3. Main Results

In this section, we compute the eccentricity based topological indices of a commutative ringZ♭1♭2♭3×Zq2, 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×Zq2, we have VJR=♭1♭2♭3q+♭1♭3q2+♭1♭2q2+♭2♭3q2+♭1q+2♭2+♭3q+q2−♭1q2−♭2q2−♭3q2−♭1♭3q−♭2♭3q+♭12q−3.Proof.
We can partition the vertex set ofR=Z♭1♭2♭3×Zq2 as follows:
(1)
ϖ1=0,v∣v∈Zq2:v≠kq,0≤k≤q−1. The vertices adjacent to x∈ϖ1 are of the form u′,0 with u′∈Z♭1♭2♭3\0. Hence, dx=♭1♭2♭3−1 for all x∈ϖ1 and ϖ1=q2−q(2)
ϖ2=0,v∣v=kq,1≤k≤q−1. So we have ϖ2=q−1. The vertices adjacent to x∈ϖ2 are of the form:
(i)
u′,0:u′∈Z♭1♭2♭3\0(ii)
u′,v′:u′∈Z♭1♭2♭3,v′=t′q,1≤t′≤q−1
Hence, for anyx∈ϖ2, dx=♭1♭2♭3−1+♭1♭2♭3−1q−1=♭1♭2♭3q−q.
(3)
ϖ3=u,0∣u=k♭1♭2;1≤k≤♭3−1 with ϖ3=♭3−1. The vertices adjacent to x∈ϖ3 are of the form:
(i)
0,v1:v1∈Zq2\0(ii)
u′,0:u′=t1♭3;1≤t1≤♭1♭2−1(iii)
u′,v2:u′=t1♭3,v2=t2q;1≤t1≤♭1♭2−1,1≤t2≤q−1(iv)
u′,v3:u′=t1♭3,v3≠t3q;1≤t1≤♭1♭2−1,1≤t3≤q−1
Hence, for anyx∈ϖ3, dx=q2−1+♭1♭2−1+♭1♭2−1q−1+♭1♭2−1q2−q=♭1♭2q2−1.
(4)
ϖ4=u,0:u=k♭1♭3;1≤k≤♭2−1. with ϖ4=♭2−1. The vertices adjacent to x∈ϖ4 are of the form:
(i)
0,v1:v1∈Zq2\0(ii)
u′,0:u′=t1♭2;1≤t1≤♭1♭3−1(iii)
u′,v2:u′=t1♭2,v2=t2q;1≤t1≤♭1♭3−1,1≤t2≤q−1(iv)
u′,v3:u′=t1♭2,v3≠t3q;1≤t1≤♭1♭3−1,1≤t3≤q−1
Hence, for anyx∈ϖ4, dx=q2−1+♭1♭3−1+♭1♭3−1q−1+♭1♭3−1q2−q=♭1♭3q2−1.
(5)
ϖ5=u,0:u=k1♭1;1≤k1≤♭2♭3−1;u≠k2♭1♭2,1≤k2≤♭3−1;u≠k3♭1♭3,1≤k3≤♭2−1 with ϖ5=♭2−1♭3−1. The vertices adjacent to x∈ϖ5 are of the form:
(i)
0,v1:v1∈Zq2\0(ii)
u′,0:u′=t1♭2♭3;1≤t1≤♭1−1(iii)
u′,v2:u′=t1♭2♭3,v2=t2q;1≤t1≤♭1−1,1≤t2≤q−1(iv)
u′,v3:u′=t1♭2♭3,v3≠t3q;1≤t1≤♭1−1,1≤t3≤q−1
Hence, for anyx∈ϖ5, dx=q2−1+♭1−1+♭1−1q−1+♭1−1q2−q=♭1q2−1.
(6)
ϖ6=u,0;u=k♭2♭3,1≤k≤♭1−1 with ϖ6=♭1−1. The vertices adjacent to x∈ϖ6 are of the form:
(i)
0,v1:v1∈Zq2\0(ii)
u′,0:u′=t1♭1;1≤t1≤♭2♭3−1(iii)
u′,v2:u′=t1♭1,v2=t2q;1≤t1≤♭2♭3−1,1≤t2≤q−1(iv)
u′,v3:u′=t1♭1,v3≠t3q;1≤t1≤♭2♭3−1,0≤t3≤q−1
Hence, for anyx∈ϖ6, dx=q2−1+♭2♭3−1+♭2♭3−1q−1+♭2♭3−1q2−q=♭2♭3q2−1.
(7)
ϖ7=u,0;u=k1♭2,1≤k1≤♭1♭3−1;u≠k2♭1♭2,1≤k2≤♭3−1;u≠k3♭2♭3,1≤k3≤♭1−1. So, ϖ7=♭1−1♭3−1. The vertices adjacent to x∈ϖ7 are of the form:
(i)
0,v1:v1∈Zq2\0(ii)
u′,0:u′=t1♭1♭3;1≤t1≤♭2−1(iii)
u′,v2:u′=t1♭1♭3,v2=t2q;1≤t1≤♭2−1,1≤t2≤q−1(iv)
u′,v3:u′=t1♭1♭3,v3≠t3q;1≤t1≤♭2−1,0≤t3≤q−1
Hence, for anyx∈ϖ7, dx=q2−1+♭2−1+♭2−1q−1+♭2−1q2−q=♭2q2−1.
(8)
ϖ8=u,0;u=k1♭3,1≤k1≤♭1♭2−1;u≠k2♭1♭31≤k2≤♭2−1;u≠k3♭2♭3,1≤k3≤♭1−1, with ϖ8=♭1−1♭2−1. The vertices adjacent to x∈ϖ8 are of the form:
(i)
0,v1:v1∈Zq2\0(ii)
u′,0:u′=t1♭1♭2;1≤t1≤♭3−1(iii)
u′,v2:u′=t1♭1♭2,v2=t2q;1≤t1≤♭3−1,1≤t2≤q−1(iv)
u′,v3:u′=t1♭1♭2,v3≠t3q;1≤t1≤♭3−1,0≤t3≤q−1
Hence, for anyx∈ϖ7, dx=q2−1+♭3−1+♭3−1q−1+♭3−1q2−q=♭3q2−1.
(9)
ϖ9=u,0;u≠k1♭1,1≤k1≤♭2♭3−1;u≠k2♭2,1≤k2≤♭1♭3−1;u≠k3♭3,1≤k3≤♭1♭2−1 with ϖ9=♭1−1♭2−1♭3−1. In this case dx=q2−1 for all x∈ϖ9(10)
ϖ10=u,v∣u=k1♭1♭2,1≤k1≤♭3−1;v≠k2q,0≤k2≤q−1 with ϖ10=♭3−1q2−q. In this case dx=♭1♭2−1 for all x∈ϖ10(11)
ϖ11=u,v∣u=k1♭1♭3,1≤k1≤♭2−1;v≠k2q,0≤k2≤q−1, with ϖ11=♭2−1q2−q. In this case dx=♭1♭3−1 for all x∈ϖ11(12)
ϖ12=u,v∣u=k1♭1,1≤k1≤♭2♭3−1;u≠k2♭1♭2,1≤k2≤♭3−1;u≠k3♭1♭3,1≤k3≤♭2−1;v≠k4q,0≤k4≤q−1 with ϖ12=♭2−1♭3−1q2−q. The vertices adjacent to x∈ϖ12 are of the form u′,0;u′=t′♭2♭3,1≤t′≤♭1−1. Hence dx=♭1−1 for all x∈ϖ12(13)
ϖ13=u,v∣u=k1♭2♭3,1≤k1≤♭1−1;v≠k2q,0≤k2≤q−1 with ϖ13=♭1−1q2−q. The vertices adjacent to x∈ϖ12 are of the form u′,0;u′=t′♭1,1≤t′≤♭2♭3−1. Hence dx=♭2♭3−1 for all x∈ϖ13(14)
ϖ14=u,v;u=k1♭2,1≤k1≤♭1♭3−1;u≠k2♭1♭2,1≤k2≤♭3−1;u≠k3♭2♭3,1≤k3≤♭1−1;v≠k4q,0≤k4≤q−1 with ϖ14=♭1−1♭3−1q2−q. Then, each vertex x=∈ϖ14 has degree ♭2−1(15)
ϖ15=u,v∣u=k1♭3,1≤k1≤♭1♭2−1;u≠k2♭1♭31≤k2≤♭2−1;u≠k3♭2♭3,1≤k3≤♭1−1;v≠k4q,0≤k4≤q−1, with ϖ15=♭1−1♭2−1q2−q. The vertices adjacent to x∈ϖ15 are of the form u′,0;u′=t′♭1♭2,1≤t′≤♭3−1. Then each vertex x=∈ϖ15 has degree ♭3−1(16)
ϖ16=u,v∣u=k1♭1♭2;1≤k1≤♭3−1;v=k2q,1≤k2≤q−1 with ϖ16=♭3−1q−1. The vertices adjacent to x∈ϖ16 are of the form:
(i)
0,v1:v1=t1q,1≤t1≤q−1(ii)
u′,0:u′=t2♭3;1≤t2≤♭1♭2−1(iii)
u′,v1:u′=t2♭3,v1=t1q;1≤t1≤q−1,1≤t2≤♭1♭2−1
Hence, for anyx∈ϖ16, dx=q−1+♭1♭2−1+♭1♭2−1q−1=♭1♭2q−1.
(17)
ϖ17=u,v∣u=k1♭1♭3,1≤k1≤♭2−1;v=k2q,1≤k2≤q−1 with ϖ17=♭2−1. The vertices adjacent to x∈ϖ17 are of the form:
(i)
0,v1:v1=t1q,1≤t1≤q−1(ii)
u′,0:u′=t2♭2,1≤t2≤♭1♭3−1(iii)
u′,v1:v1=t1q,u′=t2♭2:1≤t1≤q−1,1≤t2≤♭1♭3−1
Hence, for anyx∈ϖ17, dx=q−1+♭1♭3−1+♭1♭3−1q−1=♭1♭3q−1.
(18)
ϖ18=u,v∣u=k1♭1,1≤k1≤♭2♭3−1;u≠k2♭1♭2,1≤k2≤♭3−1;u≠k3♭1♭3,1≤k3≤♭2−1;v=k4q,1≤k4≤q−1 with ϖ18=♭2−1♭3−1q−1. The vertices adjacent to x∈ϖ18 are of the form:
(i)
u′,v′:u′=t1♭2♭3,1≤t1≤♭1−1;v′=t2q,0≤t2≤q−1(ii)
0,v1:v1=t3q,1≤t3≤q−1
Hence, for anyx∈ϖ18, dx=♭1−1q2−q+q−1=q−1♭1q−q+1.
(19)
ϖ19=u,v∣u=k1♭2♭3,1≤k1≤♭1−1;v=k2q,1≤k2≤q−1 with ϖ19=♭1−1q−1. The vertices adjacent to x∈ϖ19 are of the form:
(i)
0,v1:v1=t1q,1≤t1≤q−1(ii)
u′,0:u′=t2♭1,1≤t2≤♭2♭3−1(iii)
u′,v1:u′=t2♭1,v1=t1q;1≤t1≤q−1,1≤t2≤♭2♭3−1
Hence, for anyx∈ϖ19, dx=q−1+♭2♭3−1+♭2♭3−1q−1=♭2♭3q−1.
(20)
ϖ20=u,v∣u=k1♭2,1≤k1≤♭1♭3−1;u≠k2♭1♭2,1≤k2≤♭3−1;u≠k3♭2♭3,1≤k3≤♭1−1;v=k4q,1≤k4≤q−1 with ϖ20=♭1−1♭3−1q−1. The vertices adjacent to x∈ϖ20 are of the form:
(i)
0,v1:v1=t1q,1≤t1≤q−1(ii)
u′,0:u′=t2♭1♭3;1≤t2≤♭2−1(iii)
u′,v1:u′=t2♭1♭3,v1=t1q;1≤t1≤q−1,1≤t2≤♭2−1
Hence, for anyx∈ϖ20, dx=q−1+♭2−1+♭2−1q−1=♭2q−1.
(21)
ϖ21=u,v;u=k1♭3,1≤k1≤♭1♭2−1;u≠k2♭1♭31≤k2≤♭2−1;u≠k3♭2♭3,1≤k3≤♭11;v=k4q,1≤k4≤q−1 with ϖ21=♭1−1♭2−1q−1. The vertices adjacent to x∈ϖ21 are of the form:
(i)
0,v1:v1=t1q,1≤t1≤q−1(ii)
u′,0:u′=t2♭1♭2,1≤t2≤♭3−1(iii)
u′,v1:u′=t2♭1♭2,v1=t1q;1≤t1≤q−1,1≤t2≤♭3−1
Hence, for anyx∈ϖ21, dx=q−1+♭3−1+♭3−1q−1=♭3q−1.
(22)
ϖ22=u,v∣u≠k1♭1,1≤k1≤♭2♭3−1;u≠k2♭2,1≤k2≤♭1♭3−1;u≠k3♭3,1≤k3≤♭1♭2−1;v=k4q,1≤k4≤q−1. with ϖ22=♭1−1♭2−1♭3−1q−1. Then each vertex x=∈ϖ22 has degree q−1
Now, the cardinality of vertex setVJR can be calculated as VJR=q2−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+♭3−1q2−q+♭2−1q2−q+♭2−1♭3−1q2−q+♭1−1q2−q+♭1−1♭3−1q2−q+♭1−1♭2−1q2−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♭3q+♭1♭3q2+♭1♭2q2+♭2♭3q2+♭1q+2♭2+♭3q+q2−♭1q2−♭2q2−♭3q2−♭1♭3q−♭2♭3q+♭12q−3.
From the above Theorem, we can compute the number of edgesEJR by using hand shaking lemma.
(2)EJR=q2−q♭1♭2♭3−1+q−1♭1♭2♭3q−q+♭3−1♭1♭2q2−1+♭2−1♭1♭3q2−1+♭2−1♭3−1♭1q2−1+♭1−1♭2♭3q2−1+♭1−1♭3−1♭2q2−1+♭1−1♭2−1♭3q2−1+♭1−1♭2−1♭3−1q2−1+♭3−1q2−q♭1♭2−1+♭2−1q2−q♭1♭3−1+♭2−1♭3−1q2−q♭1−1+♭1−1q2−q♭2♭3−1+♭1−1♭3−1q2−q♭2−1+♭1−1♭2−1q2−q♭3−1+♭3−1q−1♭1♭2q−1+♭2−1♭1♭3q−1+♭2−1♭3−1q−1q−1♭1q−q+1+♭1−1q−1♭2♭3q−1+♭1−1♭3−1q−1♭2q−1+♭1−1♭2−1q−1♭3q−1+♭1−1♭2−1♭3−1q−1q−1.=2−q−1+q+q2+−2+q−1+q3+q♭2+q−2+q3+q−−6+q9+q♭2♭3+q♭1−2+q3+q−−4+q8+q♭3+♭26−q9+q+−12+q18+q♭2.
Anderson and Livingston [30] proved that the diameter of the graph JR is atmost three. It follows that the eccentricity of any vertex x∈VJR is atmost three. Figure 1 reflects this fact for the case R=Z♭1♭2♭3×Zq2.Theorem 2 (see [30]).
LetR=Z♭1♭2♭3×Zq2, where ♭1,♭2,♭3 be distinct prime integers and q is any prime number. Then, eccentricity of a vertex x∈VJR is either 2 or 3.
In the next Theorem, we find the exact expressions for theABC5, H4, GA4, M1∗M3∗ indices of R=Z♭1♭2♭3×Zq2, where ♭1,♭2,♭3 be distinct prime integers and q is any prime number. For simplicity, we fix some notations. Let α=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3,
(3)β=−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33,(4)γ=♭34+q−1+3−1+q2q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3,(5)δ=♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3,(6)ζ=−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3,(7)μ=♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3.Theorem 3.
LetR=Z♭1♭2♭3×Zq2, where ♭1,♭2,♭3 be distinct prime integers, and q is any prime number. Then, TG has the following expression TJR=αϕ2,2+β+γ+δϕ2,3+ζ+μϕ3,3,Proof.
Let(8)Ξm,n=u′v′∈EJR:εu′=m,εv′=n
Then, from Theorem1 and Figure 1, we have
(9)Ξ2,2=q−1♭3−1+q−1♭2−1+q−1p−1+♭3−1♭2−1+♭3−1♭1−1+♭2−1♭1−1,(10)Ξ2,3=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−12♭3−1+q−1♭2−1+q−12♭2−1♭3−1+q−12♭1−1+q−12♭1−1♭3−1+q−12♭1−1♭2−1+q−12♭1−1♭2−1♭3−1+♭3−1q2−q+♭1−1♭2−1♭3−1+q2−q♭2−1♭3−1+♭1−1♭3−1q2−q+♭1−1♭2−1♭3−1q2−q+♭2−1♭3−1+♭1−1♭3−1q−1+♭1−1♭2−1♭3−1q−1+♭2−1q2−q+♭1−1♭2−1♭3−1+♭2−1♭3−1q2−q+♭1−1♭2−1q2−q+♭1−1♭2−1♭3−1q2−q+♭2−1♭3−1q−1+♭1−1♭2−1q−1+♭1−1♭2−1♭3−1q−1+♭1−1q2−q+♭1−1♭2−1♭3−1+♭1−1♭3−1q2−q+♭1−1♭2−1q2−q+♭1−1♭2−1♭3−1q2−q+♭1−1♭3−1q−1+♭1−1♭2−1♭3−1q−1+♭1−1♭2−1
and(11)Ξ3,3=♭2−1♭3−1q2−q+♭1−1♭3−1q2−q+♭1−1♭2−1q2−q+4♭1−1♭2−1♭3−1q2−q+3♭1−1♭2−1♭3−1q−1+♭1−1♭2−1♭3−1+♭2−1♭3−1q−1+♭1−1♭3−1q−12+2♭1−1♭2−1♭3−1q−12+♭1−1♭2−1q−1.
Finally,(12)TJR=∑u′v′∈EJR∅εu′,εv′=∑u′v′∈Ξ2,2∅2,2+∑u′v′∈Ξ2,3∅2,3+∑u′v′∈Ξ3,3∅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∅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−12♭3−1+q−1♭2−1+q−12♭2−1♭3−1+q−12♭1−1+q−12♭1−1♭3−1+q−12♭1−1♭2−1+q−12♭1−1♭2−1♭3−1+♭3−1q2−q+♭1−1♭2−1♭3−1+q2−q♭2−1♭3−1+♭1−1♭3−1q2−q+♭1−1♭2−1♭3−1q2−q+♭2−1♭3−1+♭1−1♭3−1q−1+♭1−1♭2−1♭3−1q−1+♭2−1q2−q+♭1−1♭2−1♭3−1+♭2−1♭3−1q2−q+♭1−1♭2−1q2−q+♭1−1♭2−1♭3−1q2−q+♭2−1♭3−1q−1+♭1−1♭2−1q−1+♭1−1♭2−1♭3−1q−1+♭1−1q2−q+♭1−1♭2−1♭3−1+♭1−1♭3−1q2−q+♭1−1♭2−1q2−q+♭1−1♭2−1♭3−1q2−q+♭1−1♭3−1q−1+♭1−1♭2−1+♭1−1♭2−1♭3−1q−1∅2,3+♭2−1♭3−1q2−q+♭1−1♭3−1q2−q+♭1−1♭2−1q2−q+4♭1−1♭2−1♭3−1q2−q+3♭1−1♭2−1♭3−1q−1+♭1−1♭2−1♭3−1+♭2−1♭3−1q−1+♭1−1♭3−1q−12+2♭1−1♭2−1♭3−1q−12+♭1−1♭2−1q−1∅3,3.=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3ϕ2,2+−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33+♭34+q−1+3−1+q2q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3+♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3ϕ2,3+−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3+♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3ϕ3,3,=αϕ2,2+β+γ+δϕ2,3+ζ+μϕ3,3.Theorem 4.
LetR=Z♭1♭2♭3×Zq2, where ♭1,♭2,♭3 be distinct prime integers and q is any prime number. Then
(13)M1∗JR=4α+β+γ+δ5+ζ+μ6,(14)M3∗JR=4α+β+γ+δ6+ζ+μ9,(15)GA4JR=α+β+γ+δ265+ζ+μ,(16)ABC5JR=α2+β+γ+δ12+ζ+μ23,(17)H4JR=α2+β+γ+δ25+ζ+μ13.Proof.
For the first Zagreb eccentricity index, we have∅εu′,εv′=εu′+εv′. Then, ∅2,2=4, ∅2,3=5, and ∅3,3=6. Substituting these values in Theorem 3, we get
(18)M1∗JR=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3ϕ2,2+−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33+♭34+q−1+3−1+q−1+♭32q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3+♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3ϕ2,3+−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3+♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3ϕ3,3.=6+♭3−3+q−3q+♭2−3+♭3+q+♭1−3+♭2+♭3+q4+−4+♭12−1+♭22−1+♭33−1+q2q2+♭22−1+♭33−1+q2q2−−2+qq1+q2+♭34+−3+♭3♭3−1+q2q2+q−1+3−1+q2q+♭22q2+−2+qq2+♭3−1−2−3+♭3♭3−1+q2q2−2q1+3−1+q2q+♭13−2♭3+q−1+2−1+q2q+q−2♭22−1+♭33−1+q2q+♭3q−7−2−3+♭3♭3−1+q2−6−2+qq+♭2−2−4−1+q2q+♭3q4−3+♭3♭3−1+q2+35+4−2+qq5+−1−2−1+qq+♭3q−2+3q−♭2−2+5−4qq+♭31+5−1+qq+♭1♭3+q−2+3q+♭2−1−5−1+qq+♭3q2−4q+♭2−5+6q6,=4α+β+γ+δ5+ζ+μ6.Figure 1
Structure ofZ♭1♭2♭3×q2.For third Zagreb eccentricity index, we have∅εu′,εv′=εu′×εv′ and
(19)M3∗=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3ϕ2,2+−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33+♭34+q−1+3−1+q2q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3+♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3ϕ2,3+−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3+♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3ϕ3,3.=6+♭3−3+q−3q+♭2−3+♭3+q+♭1−3+♭2+♭3+q4+−4+♭12−1+♭22−1+♭33−1+q2q2+♭22−1+♭33−1+q2q2−−2+qq1+q2+♭34+−3+♭3♭3−1+q2q2+q−1+3−1+q2q+♭22q2+−2+qq2+♭3−1−2−3+♭3♭3−1+q2q2−2q1+3−1+q2q+♭13−2♭3+q−1+2−1+q2q+q−2♭22−1+♭33−1+q2q+♭3q−7−2−3+♭3♭3−1+q2−6−2+qq+♭2−2−4−1+q2q+♭3q4−3+♭3♭3−1+q2+35+4−2+qq6+−1−2−1+qq+♭3q−2+3q−♭2−2+5−4qq+♭31+5−1+qq+♭1♭3+q−2+3q+♭2−1−5−1+qq+♭3q2−4q+♭2−5+6q9,=4α+β+γ+δ6+ζ+μ9.For the geometric-arithmetic eccentricity index, we have∅εu′,εv′=2εu′×εv′/εu′+εv′ and
(20)GA4JR=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3ϕ2,2+−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33+♭34+q−1+3−1+q2q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3+♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3ϕ2,3+−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3+♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3ϕ3,3.=5+♭3−3+q−3q−2−1+qq+♭2−3+♭3+q+♭1−3+♭2+♭3+q+−4+♭12−1+♭22−1+♭33−1+q2q2+♭22−1+♭33−1+q2q2−−2+qq1+q2+♭34+−3+♭3♭3−1+q2q2+q−1+3−1+q2q+♭22q2+−2+qq2+♭3−1−2−3+♭3♭3−1+q2q2−2q1+3−1+q2q+♭13−2♭3+q−1+2−1+q2q+q−2♭22−1+♭33−1+q2q+♭3q−7−2−3+♭3♭3−1+q2−6−2+qq+♭2−2−4−1+q2q+♭3q4−3+♭3♭3−1+q2+35+4−2+qq265+♭3q−2+3q−♭2−2+5−4qq+♭31+5−1+qq+♭1♭3+q−2+3q+♭2−1−5−1+qq+♭3q2−4q+♭2−5+6q,=α+β+γ+δ265+ζ+μ.For atom-bond connectivity eccentricity index, we have∅εu′,εv′=εu′+εv′−2/εu′×εv′ and
(21)ABC5JR=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3ϕ2,2+−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33+♭34+q−1+3−1+q2q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3+♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3ϕ2,3+−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3+♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3ϕ3,3.=6+♭3−3+q−3q+♭2−3+♭♭3+q+♭1−3+♭2+♭3+q2+−4+♭12−1+♭22−1+♭33−1+q2q2+♭22−1+♭33−1+q2q2−−2+qq1+q2+♭34+−3+♭3♭3−1+q2q2+q−1+3−1+q2q+♭22q2+−2+qq2+♭3−1−2−3+♭3♭3−1+q2q2−2q1+3−1+q2q+♭13−2♭3+q−1+2−1+q2q+q−2♭22−1+♭33−1+q2q+♭3q−7−2−3+♭3♭3−1+q2−6−2+qq+♭2−2−4−1+q2q+♭3q4−3+♭3♭3−1+q2+35+4−2+qq12+−1−2−1+qq+♭3q−2+3q−♭2−2+5−4qq+♭31+5−1+qq+♭1♭3+q−2+3q+♭2−1−5−1+qq+♭3q2−4q+♭2−5+6q23,=α2+β+γ+δ12+ζ+μ23.For the harmonic index based on eccentricity of fourth type, we have∅εu′,εv′=2/εu′+εv′ and
(22)H4JR=6−3q+−3+q♭3+♭2−3+q+♭3+♭1−3+q+♭2+♭3ϕ2,2+−4−−2+qq1+q2+−1+q2q2♭12−1+♭22−1+♭33+−1+q2q2♭22−1+♭33+♭34+q−1+3−1+q2q+−1+q2q2−3+♭3♭3+♭22q2+−2+qq2+♭3−1−2q1+3−1+q2q−2−1+q2q2−3+♭3♭3+♭13+q−1+2−1+q2q−2♭3+q−2−1+q2q♭22−1+♭33+q♭3−7−6−2+qq−2−1+q2−3+♭3♭3+♭2−2−4−1+q2q+q♭335+4−2+qq+4−1+q2−3+♭3♭3ϕ2,3+−1−2−1+qq+q−2+3q♭3−♭2−2+5−4qq+1+5−1+qq♭3+♭1q−2+3q+−1−5−1+qq♭2+♭3+q2−4q+−5+6q♭2♭3ϕ3,3.=6+♭3−3+q−3q+♭2−3+♭3+q+♭1−3+♭2+♭3+q2+−4+♭12−1+♭22−1+♭33−1+q2q2+♭22−1+♭33−1+q2q2−−2+qq1+q2+♭34+−3+♭3♭3−1+q2q2+q−1+3−1+q2q+♭22q2+−2+qq2+♭3−1−2−3+♭3♭3−1+q2q2−2q1+3−1+q2q+♭13−2♭3+q−1+2−1+q2q+q−2♭22−1+♭33−1+q2q+♭3q−7−2−3+♭3♭3−1+q2−6−2+qq+♭2−2−4−1+q2q+♭3q4−3+♭3♭3−1+q2+35+4−2+qq25+−1−2−1+qq+♭3q−2+3q−♭2−2+5−4qq+♭31+5−1+qq+♭1♭3+q−2+3q+♭2−1−5−1+qq+♭3q2−4q+♭2−5+6q13,=α2+β+γ+δ25+ζ+μ13.


## 4. Conclusion

For zero divisor graph of commutative ringZ♭1♭2♭3×Zq2, 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 Zm×Zn for m,n∈ℤ. In future, similar work can be done for other cases of commutative ring Zm×Zn.



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