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 ring , where and are primes.

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 [19].

2. Definitions and Notations

Let and be the set of vertices and edges of connected graph , respectively. The basic notations and definitions are taken from the book [10]. Let be the degree of vertex and be the distance between two vertices and . In mathematics, eccentricity is defined as

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 [1215]. Farooq and Malik [16] introduced the “total eccentricity index” and defined as

The “first Zagreb index” (and its new version by Ghorbani and Hosseinzadeh [17]) of a graph was studied in [18] and defined as follows:

The “eccentric connectivity polynomial” [19, 20], “augmented eccentric connectivity index” [2123], “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 and . denotes the set of all vertices adjacent .

3. Results and Discussion

Beck [26] defined the zero-divisor graph as “for a commutative ring with identity and set of its all zero divisors ,” its zero-divisor graph is constructed as and , if . For further study of zero divisor, see [2732].

Let denotes the zero-divisor graph of the commutative ring . It is defined as follows: for and , if and only if , . Let , then . The vertices of the set are the nonzero divisors of the commutative ring . Also, is a nonzero divisor. Therefore, the total number of nonzero divisors is . There are total vertices of the commutative ring . Hence, there are total number of zero divisors. This implies that the order of the zero-divisor graph is , i.e., .

From the definition, we partitioned the vertex set of the graph into the following four partitions corresponding to their degrees:

This implies that . Let denotes the degree of a vertex in and denotes the distance between the vertices of two sets and . It is easy to see that , , , , , , , and . In the next theorem, we determined .

Lemma 1. The eccentricity of the vertices of is 2 or 3.

Proof. From the definition of zero-divisor graph, the vertices of the partition set are adjacent with the vertices of partitions and , i.e., . Similarly, it is observed that and , for any . Now, and . Also, the distance between any two different vertices of the same set of partition is 2. This shows that the maximum distance of the vertices of the partitions and is 2; therefore, their eccentricity is 2, i.e., .
Since partitions and are at distance 1 with partitions and , respectively, and , it follows that or . This implies that . This shows that the eccentricity of the vertices of is 2 or 3. This completes the proof.

We summarize the above discussion in Table 1.

We determined the eccentric connectivity index of in the following theorem.

Theorem 1. The eccentric connectivity index of is

Proof. Using the values from Table 1, formula (2) implies thatWe arrive at the desired result.
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 in the following corollaries.

Corollary 1. The total eccentricity index of is given by

Corollary 2. The first eccentricity Zagreb index of is given by

Theorem 2. The eccentric connectivity polynomial of is

Proof. By using the degree and its corresponding eccentricity for each partition set in Table 1, equation (5) gives

Theorem 3. The augmented eccentric connectivity index of is

Proof. Inserting values from the proof of Lemma 1 and Table 1 to equation (6), we obtainAfter simplification, we get

Theorem 4. The connective eccentric index of the graph is

Proof. We apply the values of degrees and their eccentricity from Table 1. Then, formula (7) gives

Theorem 5. The Ediz eccentric connectivity index of the graph is

Proof. By using Table 1 and equation (8), we getAfter simplification, we get

The following theorem determines the reverse eccentric connectivity index of the graph .

Theorem 6. The reverse eccentric connectivity index of the graph is

Proof. For the values given in Table 1, formula (9) implies thatAfter simplification, we getand we are done.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.


This work was supported by the Slovak Research and Development Agency under contract no. APVV-19-0153 and by VEGA 1/0233/18.