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

One of themost 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. &e computation of these topological descriptors for various chemical graphs is a very attractive direction for researchers. &e 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. &ere are some significant classes of topological descriptors and related polynomials which can be seen [1–9].


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. e computation of these topological descriptors for various chemical graphs is a very attractive direction for researchers. e 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.

Definitions and Notations
Let V(G) and E(G) be the set of vertices and edges of connected graph G, respectively. e basic notations and definitions are taken from the book [10]. Let d μ be the degree of vertex μ and d(μ, ]) be the distance between two vertices μ and ]. In mathematics, eccentricity is defined as (1) In [11], Sharma et al. introduced "eccentric connectivity index," and the general formula of eccentric connectivity index is defined as Detail of applications and results of eccentric connectivity index can be seen in [12][13][14][15]. Farooq and Malik [16] introduced the "total eccentricity index" and defined as e "first Zagreb index" (and its new version by Ghorbani and Hosseinzadeh [17]) of a graph G was studied in [18] and defined as follows: e "eccentric connectivity polynomial" [19,20], "augmented eccentric connectivity index" [21][22][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. where

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 Z(R)," its zero-divisor graph G(R) is constructed as For further study of zero divisor, see [27][28][29][30][31][32].
is implies that the order of the zero- From the definition, we partitioned the vertex set of the graph G into the following four partitions corresponding to their degrees: the degree of a vertex a in A and d(A, B) denotes the distance between the vertices of two sets A and B. It is easy to see that In the next theorem, we determined ε(Γ(Z p 2 × Z q )).

Lemma 1.
e eccentricity of the vertices of Γ(Z p 2 × Z q ) is 2 or 3.
is implies that ε(P 2 ) � ε(P 3 ) � 3. is shows that the eccentricity of the vertices of G is 2 or 3. is completes the proof.

□
We summarize the above discussion in Table 1. We determined the eccentric connectivity index of Γ(Z p 2 × Z q ) in the following theorem.

Theorem 1. e eccentric connectivity index of Γ(Z
Proof. Using the values from Table 1, formula (2) implies that We arrive at the desired result. 2 Mathematical Problems in Engineering By using Lemma 1 and Table 1 in equations (3) and (4), we obtain the total eccentricity index and the first eccentricity Zagreb index for Γ(Z p 2 × Z q ) in the following corollaries.

Corollary 1.
e total eccentricity index of Γ(Z p 2 × Z q ) is given by

Corollary 2. e first eccentricity Zagreb index of Γ(Z p 2 × Z q ) is given by
Theorem 2. e eccentric connectivity polynomial of Proof. By using the degree and its corresponding eccentricity for each partition set in Table 1, equation (5) gives Proof. Inserting values from the proof of Lemma 1 and Table 1 to equation (6), we obtain After simplification, we get □ Table 1: e summary of Γ(Z p 2 × Z q ).

Theorem 4.
e connective eccentric index of the graph Proof. We apply the values of degrees and their eccentricity from Table 1. en, formula (7) gives □ Theorem 5. e Ediz eccentric connectivity index of the graph Γ( Proof. By using Table 1 and equation (8), we get After simplification, we get □ e following theorem determines the reverse eccentric connectivity index of the graph Γ(Z p 2 × Z q ).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare no conflicts of interest.