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

Let R = Z ♭ 1 ♭ 2 ♭ 3 × Z q 2 be a commutative ring, where ♭ 1 , ♭ 2 , ♭ 3 are distinct primes, and q is any prime integer. A zero divisor graph J ð R Þ 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 J ð R Þ , namely, atom-bond connectivity index ( ABC 5 ), eccentricity-based harmonic index of fourth type ( H 4 ð J Þ ), geometric-arithmetic eccentricity index ( GA 4 ð J Þ ), eccentricity-based third Zagreb index, and eccentricity-based ﬁ rst Zagreb index.


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 countingrelated topological indices [5][6][7][8]. Atom-bond connectivity index (ABC), Randic connectivity index (R −1/2 ), geometricarithmetic 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][25][26][27][28].

Preliminary Definitions
Let G = ðV, EÞ 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 d x 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 ε x and is the maximum of the distances of all vertices from x. Mathematically ε x = max fdðx, yÞ: y ∈ Vg. 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 ∈ R are called zero divisors if x:y = 0. Let ZðRÞ be set of zero divisors of R. I. Beck [2] established the idea of the zero divisor graph JðRÞ by considering ZðRÞ as vertex set, and any two vertices x, y ∈ ZðRÞ 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 JðRÞ is a connected graph. For more results on this topic, see [24,31,32].

Main Results
In this section, we compute the eccentricity based topological indices of a commutative ring Z ♭ 1 ♭ 2 ♭ 3 × 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 JðRÞ of a ring Proof. We can partition the vertex set of R = Z ♭ 1 ♭ 2 ♭ 3 × Z q 2 as follows: The vertices adjacent to x ∈ ϖ 2 are of the form: The vertices adjacent to x ∈ ϖ 3 are of the form: The vertices adjacent to x ∈ ϖ 4 are of the form: The vertices adjacent to x ∈ ϖ 5 are of the form: The vertices adjacent to x ∈ ϖ 6 are of the form: The vertices adjacent to x ∈ ϖ 7 are of the form: The vertices adjacent to x ∈ ϖ 8 are of the form: The vertices adjacent to x ∈ ϖ 12 are of the form ðu ′ , The vertices adjacent to x ∈ ϖ 12 are of the form ðu′, 0Þ ; u′ The vertices adjacent to x ∈ ϖ 15 are of the form The vertices adjacent to x ∈ ϖ 16 are of the form: Hence, for any x ∈ ϖ 16 , The vertices adjacent to x ∈ ϖ 17 are of the form: The vertices adjacent to x ∈ ϖ 18 are of the form: The vertices adjacent to x ∈ ϖ 19 are of the form: The vertices adjacent to x ∈ ϖ 20 are of the form: The vertices adjacent to x ∈ ϖ 21 are of the form: Now, the cardinality of vertex set VðJ R Þ can be calculated as From the above Theorem, we can compute the number of edges EðJ R Þ by using hand shaking lemma.
Anderson and Livingston [30] proved that the diameter of the graph JðRÞ is atmost three. It follows that the eccentricity of any vertex x ∈ VðJðRÞÞ is atmost three. Figure 1 reflects this fact for the case Journal of Function Spaces Theorem 2 (see [30]).
be distinct prime integers and q is any prime number. Then, eccentricity of a vertex x ∈ VðJðRÞÞ is either 2 or 3.
In the next Theorem, we find the exact expressions for the where ♭ 1 , ♭ 2 , ♭ 3 be distinct prime integers and q is any prime number. For simplicity, we fix some notations. Let