# Computing Vertex-Based Eccentric Topological Descriptors of Zero-Divisor Graph Associated with Commutative Rings

Authors: Abdullah Ali H. Ahmadini; Ali N. A. Koam; Ali Ahmad; Martin Bača; Andrea Semaničová–Feňovčíková
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The applications of finite commutative ring are useful substances in robotics and programmed geometric, communication theory, and cryptography. In this paper, we study the vertex-based eccentric topological indices of a zero-divisor graphs of commutative ringℤp2×ℤq, where p and q are primes.


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## Body

## 1. Introduction

One of the most significant issues in science is to change over chemical structure into numerical molecular descriptors that are pertinent to the physical, chemical, or organic properties. Atomic structure is one of the essential ideas of science since properties and chemical and organic practices of atoms are controlled by it.Molecular descriptors called topological indices are graph invariants that play a significant job in science, and engineering, since they can be connected with huge physicochemical properties of particles. We utilize topological descriptors during the time spent associating the chemical structures with different attributes, for example, boiling points and molar heats of formation. The computation of these topological descriptors for various chemical graphs is a very attractive direction for researchers. The chemical structure of a molecule is represented by molecular descriptors.Atoms and atomic structures are frequently displayed by a molecular graph. An atomic structure is a graph in which vertices are atoms and edges are its atomic bonds. In this manner, a topological descriptor is a numeric amount related with a graph which portrays the topology of graph and its invariants. There are some significant classes of topological descriptors and related polynomials which can be seen [1–9].


## 2. Definitions and Notations

LetVG and EG be the set of vertices and edges of connected graph G, respectively. The basic notations and definitions are taken from the book [10]. Let dμ be the degree of vertex μ and dμ,ν be the distance between two vertices μ and ν. In mathematics, eccentricity is defined as(1)εμ=maxdμ,ν:ν∈VG.In [11], Sharma et al. introduced “eccentric connectivity index,” and the general formula of eccentric connectivity index is defined as(2)ξG=∑ν∈VGdνεν.Detail of applications and results of eccentric connectivity index can be seen in [12–15]. Farooq and Malik [16] introduced the “total eccentricity index” and defined as(3)ζG=∑ν∈VGεν.The “first Zagreb index” (and its new version by Ghorbani and Hosseinzadeh [17]) of a graph G was studied in [18] and defined as follows:(4)M1∗G=∑ν∈VGεv2.The “eccentric connectivity polynomial” [19, 20], “augmented eccentric connectivity index” [21–23], “connective eccentric index” [23], “Ediz eccentric connectivity index,” and “reverse eccentric connectivity index” [24, 25] are defined in equations (5)–(8) and (9), respectively.(5)ECPG,x=∑ν∈VGdvxεv,(6)ξacG=∑ν∈VGMvεv,(7)ξCG=∑ν∈VGdvεv,(8)EζG=∑ν∈VGSvεv,(9)ζReG=∑ν∈VGεvSv,where Mv=∏μ∈Nvdμ and Sv=∑μ∈Nvdμ. Nv denotes the set of all vertices adjacent ν.


## 3. Results and Discussion

Beck [26] defined the zero-divisor graph as “for a commutative ring with identity ℝ and set of its all zero divisors ℤℝ,” its zero-divisor graph Gℝ is constructed as α,β∈VGℝ=Gℝ and α,β∈EGℝ, if α.β=0. For further study of zero divisor, see [27–32].LetΓℤp2×ℤq denotes the zero-divisor graph of the commutative ring ℤp2×ℤq. It is defined as follows: for a∈ℤp2 and b∈ℤq, a,b∉VΓℤp2×ℤq if and only if a≠kp,1≤k≤p−1, b≠0. Let J=a,b∉VΓℤp2×ℤq:a≠kp,1≤k≤p−1,b≠0, then J=p2−pq−1. The vertices of the set J are the nonzero divisors of the commutative ring ℤp2×ℤq. Also, 0,0∈ℤp2×ℤq is a nonzero divisor. Therefore, the total number of nonzero divisors is J+1=p2−pq−1+1=p2q−p2−pq+p+1. There are p2q total vertices of the commutative ring ℤp2×ℤq. Hence, there are p2q−p2q−p2−pq+p+1=p2+pq−p−1 total number of zero divisors. This implies that the order of the zero-divisor graph Γℤp2×ℤq is p2+pq−p−1, i.e., VΓℤp2×ℤq=p2+pq−p−1.From the definition, we partitioned the vertex set of the graphG into the following four partitions corresponding to their degrees:(10)P1=a,b:b∈1,2,3,…,q−1,P2=a,b:a=kp,1≤k≤p−1,b∈1,2,3,…,q−1,P3=a,0:a∈ℤp2\0,p,2p,…,p−1,P4=a,0:a=kp,1≤k≤p−1.This implies thatVΓℤp2×ℤq=P1∪P2∪P3∪P4. Let dAa denotes the degree of a vertex a in A and dA,B denotes the distance between the vertices of two sets A and B. It is easy to see that P1=q−1, P2=p−1q−1, P3=p2−p, P4=p−1, dP1a=p2−1, dP2a=p−1, dP3a=q−1, and dP4a=pq−2. In the next theorem, we determined εΓℤp2×ℤq.Lemma 1.
The eccentricity of the vertices ofΓℤp2×ℤq is 2 or 3.Proof.
From the definition of zero-divisor graph, the vertices of the partition setP1 are adjacent with the vertices of partitions P3 and P4, i.e., dP1,P3=dP1,P4=1. Similarly, it is observed that dP2,P4=1 and dP4,P4=1, for any a≠b∈P4. Now, dP1,P2=dP1,P4+dP4,P2=2 and dP3,P4=dP3,P1+dP1,P4=2. Also, the distance between any two different vertices of the same set of partition P1,P2,andP3 is 2. This shows that the maximum distance of the vertices of the partitions P1 and P4 is 2; therefore, their eccentricity is 2, i.e., εP1=εP4=2.
Since partitionsP2 and P3 are at distance 1 with partitions P4 and P1, respectively, and dP1,P4=1, it follows that dP2,P3=dP2,P4+dP4,P1+dP1,P3=3 or dP2,P3=dP2,P1+dP1,P3=2+1=3. This implies that εP2=εP3=3. This shows that the eccentricity of the vertices of G is 2 or 3. This completes the proof.We summarize the above discussion in Table1.We determined the eccentric connectivity index ofΓℤp2×ℤq in the following theorem.Table 1
The summary ofΓℤp2×ℤq.
Partition sets of verticesDegreeEccentricityFrequencyP1p2−12q−1P2p−13p−1q−1P3q−13pp−1P4pq−22p−1Theorem 1.
The eccentric connectivity index ofΓℤp2×ℤq is(11)ξΓℤp2×ℤq=10p2q−8p2−11pq+5p+q+3.Proof.
Using the values from Table1, formula (2) implies that(12)ξΓℤp2×ℤq=∑ν∈VΓℤp2×ℤqdνεv=3p2−pq−1+2q−1p2−1+2p−1pq−2+3p−12q−1=10p2q−8p2−11pq+5p+q+3.
We arrive at the desired result.
By using Lemma1 and Table 1 in equations (3) and (4), we obtain the total eccentricity index and the first eccentricity Zagreb index for Γℤp2×ℤq in the following corollaries.Corollary 1.
The total eccentricity index ofΓℤp2×ℤq is given by(13)ζℤp2×ℤq=3p2+3pq−4p−q−1.Corollary 2.
The first eccentricity Zagreb index ofΓℤp2×ℤq is given by(14)M1∗ℤp2×ℤq=9p2+9pq−14p−5q+1.Theorem 2.
The eccentric connectivity polynomial ofΓℤp2×ℤq is(15)ECPΓℤp2×ℤq,x=2p2q−2p2−3pq+3p+q−1x3+2p2q−p2−pq−2p−q+3x2.Proof.
By using the degree and its corresponding eccentricity for each partition set in Table1, equation (5) gives(16)ECPΓℤp2×ℤqx=∑ν∈VΓℤp2×ℤqxdνxεv=p2−pq−1x3+p2−1q−1x2+pq−2p−1x2+q−1p−12x3=2p2q−2p2−3pq+3p+q−1x3+2p2q−p2−pq−2p−q+3x2.Theorem 3.
The augmented eccentric connectivity index ofΓℤp2×ℤq is(17)ξacΓℤp2×ℤq=p−13+q−1p2−p2pq−2p−1q−1+p3+pq−1q−12p2−1q−1p−1.Proof.
Inserting values from the proof of Lemma1 and Table 1 to equation (6), we obtain(18)ξacΓℤp2×ℤq=∑ν∈VΓℤp2×ℤqMvεv=p−1q−1pq−2p−13+q−1pq−2p−1q−1p2−p2+p2−pp2−1q−13+p−1p−1p−1q−1p2−1q−1pq−2p−22.
After simplification, we get(19)ξacΓℤp2×ℤq=p−13+q−1p2−p2pq−2p−1q−1+p3+pp−1q−1pq−2p−22p2−1q−1p−1.Theorem 4.
The connective eccentric index of the graphΓℤp2×ℤq is(20)ξCΓℤp2×ℤq=p−110pq−7p+q−76.Proof.
We apply the values of degrees and their eccentricity from Table1. Then, formula (7) gives(21)ξCΓℤp2×ℤq=∑ν∈VΓℤp2×ℤqdνεv=p2−pq−13+p2−1q−12+p−1pq−22+p−12q−13=p−110pq−7p+q−76.Theorem 5.
The Ediz eccentric connectivity index of the graphΓℤp2×ℤq is(22)EζΓℤp2×ℤq=p2+pq+p−2p−12q−13+p−13p2q+2pq2−2p2−7pq+p−2q+62.Proof.
By using Table1 and equation (8), we get(23)EζΓℤp2×ℤq=∑ν∈VΓℤp2×ℤqSvεv=p−12q−1pq−23+p−13q−1+p−1p2−1q−1+p−2pq−22+p−1pq−2q−1+q−12p2−p2+p2−pq−1p2−13.
After simplification, we get(24)EζΓℤp2×ℤq=p2+pq+p−2p−12q−13+p−13p2q+2pq2−2p2−7pq+p−2q+62.The following theorem determines the reverse eccentric connectivity index of the graphΓℤp2×ℤq.Theorem 6.
The reverse eccentric connectivity index of the graphΓℤp2×ℤq is(25)ζReΓℤp2×ℤ=3p2+pq+p−2pp+1p−12q−1pq−2+2pp−12q−1+p−2pq−2+2p−1q−12pq−p−2.Proof.
For the values given in Table1, formula (9) implies that(26)ζReΓℤp2×ℤq=∑ν∈VΓℤp2×ℤqεvSv=3p−12q−1pq−2+2p−13q−1+p−1p2−1q−1+p−2pq−2+2p−1pq−2q−1+q−12p2−p+3p3−pq−1p2−1.
After simplification, we get(27)ζReΓℤp2×ℤq=3p2+pq+p−2pp+1p−12q−1pq−2+2pp−12q−1+p−2pq−2+2p−1q−12pq−p−2,and we are done.



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