Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

Let R be a commutative ring with nonzero identity and let Z(R) be its set of zero divisors. )e zero-divisor graph of R is the graph Γ(R) with vertex set V(Γ(R)) � Z(R)∗, where Z(R)∗ � Z(R)\ 0 { }, and edge set E(Γ(R)) � x, y 􏼈 􏼉: x · y � 0 􏼈 􏼉. One of the basic results for these graphs is that Γ(R) is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is Zp2q2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.


Introduction
Algebraic structures have been investigated significantly for their nearby connection with representation theory and number theory; likewise, they have been widely concentrated in combinatorics [1,2]. Despite the expansive theoretical research in these areas, restricted rings and fields got consideration for their applications to cryptography and coding theory.
In mathematical chemistry, a graphical structure of a chemical compound is a representation of the structural formula. In a chemical graph, vertices and edges represent the atoms and their chemical bonds of the compound, respectively. Molecular descriptors for a particular chemical compound are calculated on basis of the corresponding molecular graph. A topological index is a graph invariant that is obtained from it. In [3], the first topological index, namely, the Wiener index, was introduced. Nowadays, it is widely used in QSAR ("Quantitative Structure Activity Relationship"), whose properties are surveyed in [4,5].
Topological indices are classified as degree based [6][7][8][9] and distance based of graphs. Some well-known topological indices based on the degrees of a graph are the Randić connectivity index, Zagreb indices, Harmonic index, atom bond connectivity, and geometric arithmetic index. e Wiener index, Hosaya index, and Estrada index are distance-based topological indices [10,11]. Topological indices formulate the criteria for the development of compound structures, and numerical activities on these structures extend multidisciplinary research. In what follows, we cite some of them.
A relationship among the stability of linear alkanes and the branched alkanes is examined using the ABC index, which helped in computation of strain energy for cycle alkanes [12,13]. e GA index is more appropriate and efficient to correlate certain physico-chemical characteristics for predictive power than the Randić connectivity index [14,15]. e Zagreb indices are powerful tools for the calculation of total p-electron energy of the molecules with precise approximation [16]. e degree-based topological indices are more useful to examine the chemical characteristics of distinct molecular structures. Eccentricity-based topological indices are useful as a key for the judgement of toxicological, physicochemical, and pharmacological properties of a compound through the structure of its molecules. e study of the QSAR is known for this sort of analysis [17]. By exploring [18,19], further applications of topological indices can be obtained. this paper. We recall some concepts from graph theory. Let G be a (undirected) graph. If there is a path between any two distinct vertices of G, then G is a connected graph. For two distinct vertices x, y ∈ V(G), we denote d(x, y) the length of a shortest path connecting x and y (d(x, x) � 0 and, d(x, y) � ∞ if no such a path exists). e diameter of the graph G is the maximum length of the shortest path con- e number of edges incidence a vertex x of simple graph G is called the degree of the vertex x, denoted as d x . e Wiener index [11] was introduced by Wiener in 1947 to illustrate the connection between physico-chemical properties of organic compounds and the index of their molecular graphs: Randić [20] and Randić et al. [21] introduced a modified version of the Wiener index that is used for predicting physico-chemical properties of organic components. e new index was called the hyper-Wiener index and it is defined as follows: e Hosoya polynomial was introduced in 1989 [22]. e definition is as follows: Dobrynin and Kochetova [23], and independently, Gutman [24] introduced a degree distance index, which is known as the Schultz index. Let G be a connected graph and d u be the degree of u ∈ V(G). en, the Schultz index or the degree distance of G is defined as follows: Klavžar and Gutman defined, in [25], the modified Schulz index of a graph as follows: Finally, Gutman, in [24], introduced two topological polynomials, namely, the Schulz polynomial Sc(G, x) and the modified Schulz polynomial Sc * (G, x) as follows: e connection between the above polynomials and the previous two indices is stated below: 1.2. Zero-Divisor Graphs. Let R be a commutative ring with nonzero identity and let Z(R) be its set of zero divisors. e zero-divisor graph of R is the graph As usual, an edge x, y is simply denoted as xy. Zero-divisor graphs were introduced by Beck [2] in 1988 and then studied by Anderson and Naseer in [26]. ese authors were interested in colorings and the original definition included all elements in R, even the zero. Later on, Anderson and Livingston [27] made emphasis on the relationship between ring-theoretical properties and graph-theoretical properties and reformulated the definition as it appears in the lines above. One of the basic results in this relationship is the following one.
Theorem 1 (see [27]). Let R be a commutative ring. en, Γ(R) is connected with diameter less or equal to 3. e study conducted in [28,29] may serve as a survey that is very interesting to find the relation between ringtheoretic properties and graph-theoretic properties of Γ(G). Some applications and relation between algebraic theory and chemical graph theory can be seen in [1,18,30]. In this paper, we presented some results that interplay in the relation between a zero-divisor graph and chemical graph theory. e structure of the paper is as follows. In Section 2, we describe the family of zero-divisor graphs of the form Γ(Z p 2 q 2 ), where p and q are different primes, and we also count pairs of vertices that are exactly at distance i, for i � 1, 2, 3. In Section 3, we obtain the Wiener index and the Hosoya, the Shultz, and the modified Shultz polynomials of Γ(Z p 2 q 2 ). We also obtain the Shultz and the modified Shultz indices of Γ(Z p 2 q 2 ).

The Zero-Divisor Graph on Γ(Z p 2 q 2 )
Let us start by introducing some notation that will be used along the paper. We assume that p and q are different positive primes.
e vertices of Γ(Z p 2 q 2 ) can be split into blocks such that all vertices in the same block have the same behavior. From this partition, we can easily describe the structure of Γ(Z p 2 q 2 ), that is, the content of the following lemma. Moreover, According to the definition, all vertices in the same block of the zero-divisor graph on Z p 2 q 2 have the same degree. For more details on this graph, see [30]. Let d ij be the degree of any vertex in B i,j . Following the notation of the previous lemma, we also conclude the following information. Figure 1 shows the structure of the zero-divisor graph Z p 2 q 2 . White vertices represent blocks of independent vertices in Z p 2 q 2 , whereas black vertices represent cliques. e structure shown in Lemmas 2 and 3 can be completed by showing the distance between pairs of vertices, which only depends on the block they belong to. is information appears in Table 1.
Let TP i (G), i ∈ Z and i > 0, the number of pairs of vertices at distance i in a graph G. Lemma 4. Let Γ(Z p 2 q 2 ) be a zero-divisor graph; then, Proof. e size of Γ(Z p 2 q 2 ) is given by (11) at is, by introducing the values described in Lemma 2, the result follows. □ Lemma 5. Let Γ(Z p 2 q 2 ) be a zero-divisor graph; then, Proof. e number of pairs of vertices at distance 2 is given by the formula that follows: that is, us, by Lemma 2, we obtain the following expression: Hence, after simplification, the result follows. □ Lemma 6. Let Γ(Z p 2 q 2 ) be a zero-divisor graph; then, Figure 1: e structure of Z p 2 q 2 .
Proof. e number of pairs at distance exactly 3, TP 3 , is given by the following expression: us, by Lemma 2, we get the result.

Distance-Based Topological Indices and Polynomials of Γ(Z p 2 q 2 )
Now, we are ready to state and prove the following theorems.
Proof. e diameter of Γ(Z p 2 q 2 ) is 3. us, there are pairs of vertices at distance 1, 2, and 3, and the Winner index can be obtained as follows: By Lemmas 4-6, we get TP 1 , TP 2 , and TP 3 , respectively. us, by introducing these values in (19), we get, after simplification, the required result.

Theorem 3. e Hosoya polynomial of Γ(Z
Proof. e result follows by Lemma 7 and by Lemmas 4-6.
Mathematical Problems in Engineering Theorem 6. Let p and q be different primes.

Conclusion
e structure of zero-divisor graphs of the form of Γ(Z p 2 q 2 ) is particularly interesting for studying distancebased topological indices. First, because its diameter is exactly 3, but also because there are defined blocks, with a complete bipartite connection between them, of either independent vertices or complete graphs (cliques). In this paper, we have focused on the Wiener index and the Hosoya, the Shultz, and the modified Shultz polynomials of Γ(Z p 2 q 2 ) and, finally, on the Shultz and the modified Shultz indices of Γ(Z p 2 q 2 ). For that reason, we have introduced some notation that could be useful not only for studying other distance-base topological indices of Γ(Z p 2 q 2 ) but also for other graphs of the form Γ(Z p m q n ), for m, n positive integers. A key point in this notation is the study of pairs that are exactly a distance one (the size of the graph), two, or three. We think that a possible line of future research should include the study of paths connecting pairs of vertices a different distances and the extension to other indices, as for instance, the Estrada index.

Data Availability
All the data are provided within the manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interest.
Mathematical Problems in Engineering 7