Topological Indices of Total Graph and Zero Divisor Graph of Commutative Ring: A Polynomial Approach

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Introduction
Troughout this article, we consider connected graph containing no loops or parallel edges.Let V(G) and E(G) be the vertex set and the edge set of a graph G, respectively.Te degree of a vertex v ∈ V(G), denoted by d v , is the total number of edges incident on v. Te distance between two nodes u, v is the length of shortest path between them and it is denoted by d(u, v).In mathematical chemistry, topological indices play a key role in predicting diferent physicochemical properties, chemical reactivity, and biological activities.Te topological index is a mapping from the collection of all graphs to the set of real numbers that yields the same value for isomorphic graphs.Te Wiener index was the frst topological index presented by Wiener [1] in 1947 to predict the boiling point of parafn.Te Wiener index is a distancebased index, which is defned for a graph G as follows: d (u, v). ( After that, a large number of topological indices have been developed in the feld of chemical graph theory to develop diferent structure-property/structure activity models.Te algebraic polynomial plays signifcant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices.Defnitions are commonly used to calculate topological descriptors.An efective general strategy [2,3] for generating topological indices of a specifc class is to compute the graph polynomial frst and then take the integral, derivative, or both of the graph polynomial at a specifc point.For instance, the Hosoya polynomial [4] is a general polynomial in the feld of distance-based topological indices whose derivatives at 1 yield the Wiener index [3].Te Hosoya polynomial of a graph G is formulated as follows: where d(G, κ) is the number of pairs of nodes (u, v) having distance κ in G and D � max d(u, v): u, v ∈ V(G) { }.It is clear that d(G, 0) and d(G, 1) are the order (total number nodes) and size (total number edges) of G, respectively.Te frst derivative of the Hosoya polynomial at x � 1 yields the Wiener index as follows: Te hyper-Wiener index is another distance-based index which was introduced by Randić in 1993 [5] for the study of structure-property relationships of molecule and its formulation is as follows: Cash et al. [6] established that the hyper-Wiener index can also be obtained from Hosoya polynomial which is given as follows: dx 2          x�1 . ( Diferent versions of Wiener indices for symmetric molecular structures were computed in [7].In a recent work, Peng et al. [8] derived Wiener and hyper-Wiener indices of polygonal cylinder and torus.In 1989, Schultz introduced a topological index known as Schultz index [9] which is a blend of degree and distance.Te Schultz index of a graph G is defned by Later, in 1997, Klavzar and Gutman presented another degree-distance based indices known as the modifed Shultz index [10] which is defned as follows: Tere are two polynomials in the literature introduced by Gutman in 1994 [11] known as Shultz and modifed Shultz polynomials that can be used to compute the indices described in equations ( 6) and (7).Te Schultz polynomial of a graph G is defned by Te modifed Schultz polynomial of a graph G is formulated as follows: Te relations between the Shultz and the modifed Shultz indices with their corresponding polynomials are as follows: In the area of degree-based topological indices, Mpolynomial [2] performs a similar role to compute closed expressions of many degree based topological indices.Te M-polynomial of a graph G is defned as follows: where We use M(G) for M(G; x, y) in this article.Any degree-based topological indices for a graph G can be expressed as follows: where f � f(x, y) is a function appropriately selected for possible chemical applications [12].Te above result can also be written as Gutman and Trinajstić introduced Zagreb indices [13] in 1972.Te frst Zagreb index is defned as follows: Te second Zagreb index is defned as follows: Furtula and Gutman [14] defned forgotten topological index in 2015, which is formulated as follows: Nikolic et al. [15] introduced the second modifed Zagreb index in 2003 which is defned by Bollobas and Erdos [16] and Amic et al. [17] presented the idea of the generalized Randić index and discussed it widely in both chemistry and mathematics [18].For more discussion, readers are referred [19][20][21].Te general Randić index is defned as follows: Te inverse Randić index is defned as follows: In 2010, Vukicevic [22] presented the symmetric division deg index of a connected graph G that is defned by Te harmonic index [23] of a graph was defned in 1987 by Fajtlowicz.It is defned by In 1982, Balaban [24] presented the inverse sum indeg index as follows: Te augmented Zagreb index was proposed by Furtula et al. [25] in 2010 that is defned as follows: Te relations of some degree-based topological indices with the M-polynomial are shown in Table 1. Here, Tus, computation of degree-based topological indices reduces to the evaluation of a single polynomial.Moreover, detailed analysis of this polynomial can yield new insights into the knowledge of degree-based topological indices.Munir et al. computed degree-based topological indices of nanostar dendrimers [26], titania nanotubes [27], and polyhex nanotubes [28] using the M-polynomial approach.In [29], Mpolynomials and topological indices of V-phenylenic nanotubes and nanotori were evaluated.For some current works on M-polynomials, readers are referred to [30][31][32].M-Polynomials of circulant graphs were studied in [33].
Algebraic structures have been extensively studied in combinatorics due to their close relationship with representation theory and number theory [34,35].Tree decades ago, a strong association between commutative algebra and graph theory was established by means of a zero divisor graph.
Graphs associated to fnite commutative rings is a fastdeveloping area and are seen to have important applications in numerous areas like algebraic cryptography and coding theory, information and communication theory, and so on [36][37][38][39].Te input model of a frequency assignment network is a zero-divisor graph of commutative ring [40].Let Z(R) and U(R) be the set of zero divisors and the set of units, respectively, of a commutative ring R with unity [41].Beck [35] introduced a connection between a commutative ring and a graph.He defned a graph taking the elements of R as vertices and two elements u, v ∈ R are connected if uv � 0. Ten Anderson and Livington [42] introduced the zero divisor graph Γ(R), where the nonzero zero-divisors are taken as vertices and two vertices u, v are adjacent if uv � 0. In [43], Anderson and Badawi introduced total graph of a commutative ring with unity.Te total graph of R, denoted by T(R), is formed by considering the elements of R as vertices and two vertices u, v ∈ V(T(R)) are connected by an edge if u + v ∈ Z(R).A graph invariant is an attribute of a graph that remains unchanged under graph isomorphism.Topological indices are considered to be the numerical graph invariants that can model the structural features of a graph.Tus, the derivation of such invariants for structures defned on a fnite commutative ring is benefcial from both theoretical and applied points of view.Nikmehr et al. [44] computed some degree, and distance-based indices of the total graph of the commutative ring Z n , where n ∈ Z + .In [45], Akgunes and Nacaroglu computed some degree-based topological indices of zero-divisor graphs generated by commutative rings.In a recent work [39], Elahi et al. [39] derived some eccentricity-based indices of zero-divisor graphs.For some current works on computations of topological indices for the total graph and the zero divisor graph of a commutative ring, readers are referred to [46][47][48].Te goal of the present work is to obtain diferent distance-based, degreedistance-based, and degree-based topological indices of the total graph T(Z n ) (n ∈ Z + ), the zero divisor graph Γ(Z r n ) (r is prime, n ∈ Z + ), and the zero divisor graph Γ(Z r × Z s × Z t ) (r, s, t are primes) using Hosoya polynomial, Scultz polynomial, modifed Scultz polynomial, and M-polynomial.

Motivation
Te journey of topological indices was started through the Wiener index [1] in 1947, where the boiling points of the parafns were modelled as follows: where t B is the boiling point; w, p are the Wiener index and polarity number respectively; and a, b, and c are constants for a given isomeric group.Te quantitative structure-property relationships were established between boiling points and hyper-Wiener index in a series of cyclic and acyclic alkanes [49].Te frst and second Zagreb indices were shown to be efective in the computation of the total π-electron energy of molecule [50].Tey were suggested for the approximation of stretched carbon-skeleton [13].Te linear combination of the forgotten topological index and the frst Zagreb index gives a mathematical model of certain physico-chemical properties of alkanes with high accuracy [14].Randić observed the correlation between the Randic index and physio-chemical properties of alkane such as boiling point, enthalpy of formation, surface area, and so on.Te strategy of encoding information on the molecular structure using topological indices has a low computational cost and a high predictive potential.Furthermore, these molecular descriptors provide insights about structural characteristics that are easily identifed.Te study of graphs built from commutative rings focuses on the interaction between the algebraic and graph theoretical properties of the associated graph.Tis connection is applicable to information in communication theory.It is therefore worthwhile to compute the topological indices of the total graph and the zero divisor graph of the commutative ring.

Methodology
Our main fndings include topological indices of the total graph and the zero divisor graph of the commutative ring using an algebraic polynomial approach.We consider the total graph , and the zero divisor graph Γ(Z r × Z s × Z t ) (r, s, t are primes).To get our outcome, we utilize the method of combinatorial derivation, vertex, and edge partition method, graph theoretical tools, analytic techniques, degree, and distance counting methods.Firstly we obtain the Hosoya polynomial, Schultz polynomial, modifed Schultz polynomial, and the Mpolynomial.Using these polynomials, we derive degree based, degree-distance based, and distance based indices.

Main Results
In this section, we give our main computational results and divide the section into three sections.

Total Graph T(Z n ). Troughout this section, the ring of integers modulo n
Let the number of zero divisors of Z n be m.We start with the following lemma.
Observation 1 (see [51]).Te degree of any vertex Observation 2 (see [44]).Te cardinality of the edge set E(T(Z n )) is as follows: Now, we compute the M-polynomial of the total graph Theorem 1.Let G be the total graph T(Z n ).Ten, we have Proof.Te edge set of T(Z n ) can be partitioned as follows: where Also, ]. From the defnition, the M-polynomial of G is obtained as follows: Now, Tis completes the proof.Now, using this M-polynomial, we calculate some degree based topological index of the total graph T(Z n ) in the following theorem.

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Theorem 2. Let G be the total graph T(Z n ).Ten, we have Ten, we have 1 ) is odd.

􏽮 .
Using Table 1, we can easily obtain the required result.

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Example 1.Using Teorem 2, we obtain the degree based topological indices of the total graph T(Z n ) for n � 6, 10, 15 in Table 2.

Zero Divisor Graph Γ(Z r n ).
In this section, we obtain some distance-and degree-distance-based indices of the zero divisor graph Γ(Z r n ), where r is prime and n is natural number.

Theorem 3. Let G be the zero divisor graph Γ(Z r n ). Ten, the Hosoya polynomial of G is given by
where and Proof.Te collection of all non-zero zero divisors of Z r n is as follows: We can partition the set Z * (Z r n ) as follows: where 6 Complexity Te rest of the proof can be constructed by considering the following two cases.
Case 1. n is even.All the nodes belong to the sets P n/2 , P n/2+1 , . . ., P n− 1 are mutually connected.Total number of such nodes is r n/2 − 1. Tus total number of such edges is (r n/2 − 1)(r n/2 − 2)/2 � I 11d .Each vertex of P i is adjacent to every vertex of P n− 1 , P n− 2 , . . ., P n− i and we denote this adjacency by P i − (P n− 1 , P n− 2 , . . ., P n− i ), where i � 1, 2, . . ., n/2 − 1.We describe such connections in Table 3.Let I 12 be the total number of edges described in Table 3. Ten, we have Tus, the total number of edges of Γ(Z r n ) is No two nodes belong to the partitions P 1 , P 2 , . . ., P n/2− 1 are connected.From P 1 to P n/2− 1 , each pair of nodes have distance 2. Total number of such pairs is (r  4. Let I 22 be the total number of edges described in Table 4. Ten, we have Table 3: Details of edge connections.

Edge connection Number of edges
8 Complexity Tus, total number of pairs (u, v) such that d(u, v) � 2 is I 21 + I 22 � I 2 .Tus, using equation ( 2), we obtain the Hosoya polynomial of G as follows: Case 2. n is odd.
All the nodes belong to the sets P n+1/2 , P n/2+1 , . . ., P n− 1 are mutually connected.Total number of such nodes is r n− 1/2 − 1. Tus total number of such edges is (r n− 1/2 − 1)(r n− 1/2 − 2)/2 � J 11 , say.Each vertex of P i is adjacent to each vertex of P n− 1 , P n− 2 , . . ., P n− i , where i � 1, 2, . . ., n − 1/2.Similar to the frst case, we can say that the total number of edges of Γ(Z r n ) is J 11 + J 12 , where It is clear that J 11 + J 12 � J 1 .No two nodes belong to the partitions P 1 , P 2 , . . ., P n− 1/2 are connected.From P 1 to P n− 1/2 , each pair of nodes have distance 2. Total number of such pairs is (r It is obvious that J 21 + J 22 � J 2 .Tus, using equation (2), we obtain the Hosoya polynomial of G as follows: Using equation ( 3) and Teorem 3, we obtain the Wiener index of Γ(Z r n ) in Teorem 4.

□ Theorem 4. Let G be the zero divisor graph Γ(Z r n ). Ten, the Wiener index of G is given by
where I 1 , I 2 , J 1 , and J 2 are given in Teorem 3.

Theorem 5. Let G be the zero divisor graph Γ(Z r n ). Ten, the hyper-Wiener index of G is given by
where I 1 , I 2 , J 1 , and J 2 are given in Teorem 3.

Theorem 6. Let G be the zero divisor graph Γ(Z r n ). Ten, the Schultz polynomial of G is given by
n is even, Complexity 9 where L i , L i ′ , Q i , and Q i ′ , i � 1, 2, 3, are described in proof.
Proof.To construct the proof, we consider the partition of Γ(Z r n ) as illustrated in the proof of Teorem 3. Te degrees of vertices G are as follows: , n is even, , n is even, , n is odd, where u i ∈ P i , i � 1, 2, . . ., n − 1.We consider the following two cases.
Case 1. n is even.First, we consider the partitions of all pairs (u, v) having distance 1.We know that d(u, v) � 1 for all u, v ∈ P i , i � n/2, n/2 + 1, . . ., n − 1.We describe the partitions of all such pairs (u, v) in Table 5. Let We describe the partitions of all such pairs (u, v) in Table 6.
We describe the partitions of all such pairs (u, v) in Table 7.
. Now, we consider the partitions of all pairs (u, v) having distance 2. We have d(u, v) � 2 for all u, v ∈ P i , i � 1, 2, . . ., n/2 − 1.We describe the partitions of all such pairs (u, v) in Table 8.
We describe the partitions of all such pairs (u, v) in Table 9.
We describe the partitions of all such pairs (u, v) in Table 10.
First, we consider the partitions of all pairs (u, v) with d(u, v) � 1.We know that d(u, v) � 1 for all u, v ∈ P i , i � n + 1/2, n + 1/2 + 1, . . ., n − 1.We describe the partitions of all such pairs (u, v) in Table 11. Let We describe the partitions of all such pairs (u, v) in Table 12. Let We describe the partitions of all such pairs (u, v) in Table 13. Let . Now, we consider the partitions of all pairs (u, v) having distance 2. We have d(u, v) � 2 for all u, v ∈ P i , i � 1, 2, . . ., n − 1/2.We describe the partitions of all such pairs (u, v) in Table 14. Let We describe the partitions of all such pairs (u, v) in Table 15. Let We describe the partitions of all such pairs (u, v) in Table 16.

□ Theorem 7. Let G be the zero divisor graph Γ(Z r n ). Ten, the Schultz index of G is given by
, are described in proof of Teorem 6.

Theorem 8. Let G be the zero divisor graph Γ(Z r n ). Ten, the modifed Schultz polynomial of G is given by
n is even, where Using equation (11) and Teorem 8, we obtain the modifed Schultz index of Γ(Z r n ) in Teorem 9.

Theorem 9. Let G be the zero divisor graph Γ(Z r n ). Ten, the modifed Schultz index of G is given by
where L i , L i ′ , Q i , and Q i ′ , i � 1, 2, 3, are described in Teorem 8.

Zero Divisor
In this section, we obtain some distance-based, degree-distance-based, and degree-based indices of the zero divisor graph Γ(Z r × Z s × Z t ), where r, s, t are primes.Te structure of the zero divisor graph Γ(Z 5 × Z 3 × Z 2 ) is shown in Figure 1.We start by computing the Hosoya polynomial in Teorem 10.
Theorem 10.Let G be the zero divisor graph Γ(Z r × Z s × Z t ).Ten, its Hosoya polynomial is given by where A α is the number of pair of nodes having distance α.
Proof.Te number of nodes and edges of G are rs . Now, we fnd A 2 and A 3 .We list the information about all pairs of nodes of G having distance 2 in Table 17.Tus, we have A 2 � summation of all the numbers in the second column of Table 17.Now, we consider the pair of nodes having distance 3 in Table 18.
Tus, we have A 3 � summation of all the numbers in the second column of Table 18.Now, using equation (2), we can compute the required result.Now, using equations (3) and ( 5), we can obtain the Wiener and hyper-Wiener indices of Γ(Z r × Z s × Z t ) in Teorem 11.

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Theorem 11.Let G be the zero divisor graph Γ(Z r × Z s × Z t ).Ten, we have where A α is the number of pair of nodes having distance α.
Now we are going to get the M-polynomial.For this, we need to know the degree of all nodes, which is illustrated in Lemma 1.

Now,
Table 17: Information related to the pair of nodes of G having distance 2.

Conclusion
We obtained some degree-based, degree-distance-based, and distance-based topological indices of the total graph T(Z n ) (n ∈ Z + ), the zero divisor graph Γ(Z r n ) (r is prime, n ∈ Z + ), and the zero divisor graph Γ(Z r × Z s × Z t ) (r, s, t are primes) through algebraic polynomials.Firstly, we computed the explicit expressions of algebraic polynomials for the aforesaid structures.Ten, topological indices are recovered from the polynomials.Te results can play an important role in visualizing the topology of the graphs under consideration.
, and c kl ′ are illustrated in the proof of Teorem 6.

Table 1 :
Derivation of some degree-based topological indices.

Table 2 :
Topological indices for the total graph T(Z

Table 4 :
Details of pairs having distance 2.

Table 15 :
Partition of pair of nodes (u, v) having distance 2.