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

Authors: Sourav Mondal; Muhammad Imran; Nilanjan De; Anita Pal
Journal: Complexity
 (2023)

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## Abstract

The algebraic polynomial plays a significant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices. The topological index is utilized as a significant tool in the study of the quantitative structure activity relationship (QSAR) and quantitative structures property relationship (QSPR) which correlate a molecular structure to its different properties and activities. Graphs containing finite commutative rings have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this article, the topological indices of the total graphTℤnn∈ℤ+, the zero divisor graph Γℤrn (r is prime, n∈ℤ+), and the zero divisor graph Γℤr×ℤs×ℤt (r,s,t are primes) are computed using some algebraic polynomials.


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## Body

## 1. Introduction

Throughout this article, we consider connected graph containing no loops or parallel edges. LetVG and EG be the vertex set and the edge set of a graph G, respectively. The degree of a vertex v∈VG, denoted by dv, is the total number of edges incident on v. The distance between two nodes u,v is the length of shortest path between them and it is denoted by du,v. In mathematical chemistry, topological indices play a key role in predicting different physico-chemical properties, chemical reactivity, and biological activities. The 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. The Wiener index was the first topological index presented by Wiener [1] in 1947 to predict the boiling point of paraffin. The Wiener index is a distance-based index, which is defined for a graph G as follows:(1)WG=∑u,v⊆VGdu,v.After that, a large number of topological indices have been developed in the field of chemical graph theory to develop different structure-property/structure activity models. The algebraic polynomial plays significant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices. Definitions are commonly used to calculate topological descriptors. An effective general strategy [2, 3] for generating topological indices of a specific class is to compute the graph polynomial first and then take the integral, derivative, or both of the graph polynomial at a specific point. For instance, the Hosoya polynomial [4] is a general polynomial in the field of distance-based topological indices whose derivatives at 1 yield the Wiener index [3]. The Hosoya polynomial of a graph G is formulated as follows:(2)HG,x=∑κ=0DdG,κxκ,where dG,κ is the number of pairs of nodes u,v having distance κ in G and D=maxdu,v:u,v∈VG. It is clear that dG,0 and dG,1 are the order (total number nodes) and size (total number edges) of G, respectively. The first derivative of the Hosoya polynomial at x=1 yields the Wiener index as follows:(3)WG=dHG,xdx|x=1.The 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:(4)WWG=12∑u,v⊆VGdu,v+du,v2.Cash et al. [6] established that the hyper-Wiener index can also be obtained from Hosoya polynomial which is given as follows:(5)WWG=dHG,xdx|x=1+12d2HG,xdx2|x=1.Different 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. The Schultz index of a graph G is defined by(6)ScG=∑u,v⊆VGdu+dvdu,v.Later, in 1997, Klavzar and Gutman presented another degree-distance based indices known as the modified Shultz index [10] which is defined as follows:(7)Sc∗G=∑u,v⊆VGdudvdu,v.There are two polynomials in the literature introduced by Gutman in 1994 [11] known as Shultz and modified Shultz polynomials that can be used to compute the indices described in equations (6) and (7). The Schultz polynomial of a graph G is defined by(8)ScG,x=∑u,v⊆VGdu+dvxdu,v.The modified Schultz polynomial of a graphG is formulated as follows:(9)Sc∗G,x=∑u,v⊆VGdudvxdu,v.The relations between the Shultz and the modified Shultz indices with their corresponding polynomials are as follows:(10)ScG=dScG,xdx|x=1,(11)Sc∗G=dSc∗G,xdx|x=1.In the area of degree-based topological indices, M-polynomial [2] performs a similar role to compute closed expressions of many degree based topological indices. The M-polynomial of a graph G is defined as follows:(12)MG;x,y=∑i⩽jMi,jGxiyj,where Mi,jG is the total count of edges uv∈EG such that du,dv=i,j. We use MG for MG;x,y in this article. Any degree-based topological indices for a graph G can be expressed as follows:(13)IG=∑uv∈EGfdu,dv,where f=fx,y is a function appropriately selected for possible chemical applications [12]. The above result can also be written as(14)IG=∑i⩽jMi,jGfi,j.Gutman and Trinajstić introduced Zagreb indices [13] in 1972. The first Zagreb index is defined as follows:(15)M1G=∑v∈VGdv2.The second Zagreb index is defined as follows:(16)M2G=∑uv∈EGdudv.Furtula and Gutman [14] defined forgotten topological index in 2015, which is formulated as follows:(17)FG=∑uv∈EGdu2+dv2.Nikolic et al. [15] introduced the second modified Zagreb index in 2003 which is defined by(18)M2mG=∑uv∈EG1dudv.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 [19–21]. The general Randić index is defined as follows:(19)RαG=∑uv∈EGdudvα.The inverse Randić index is defined as follows:(20)RRαG=∑uv∈EG1dudvα.In 2010, Vukicevic [22] presented the symmetric division deg index of a connected graph G that is defined by(21)SDDG=∑uv∈EGmindu,dvmaxdu,dv+maxdu,dvmindu,dv.The harmonic index [23] of a graph was defined in 1987 by Fajtlowicz. It is defined by(22)HG=∑uv∈EG2du+dv.In 1982, Balaban [24] presented the inverse sum indeg index as follows:(23)IG=∑uv∈EGdudvdu+dv.The augmented Zagreb index was proposed by Furtula et al. [25] in 2010 that is defined as follows:(24)AG=∑uv∈EGdudvdu+dv−23.The relations of some degree-based topological indices with the M-polynomial are shown in Table1.Table 1
Derivation of some degree-based topological indices.
Topological indexf(x, y)Derivation fromMGM1x+yDx+DyMG|x=y=1M2xyDxDyMG|x=y=1Fx2+y2Dx2+Dy2MG|x=y=1mM21/xySxSyMG|x=y=1RαxyαDxαDyαMG|x=y=1RRα1/xyαSxαSyαMG|x=y=1SDDx2+y2/xyDxSy+SxDyMG|x=y=1H2/x+y2SxJMG|x=1Ixy/x+ySxJDxDyMG|x=1Axy/x+y−23Sx3Q−2JDx3Dy3MG|x=1Here,(25)Dxfx,y=x∂fx,y∂x,Dyfx,y=y∂fx,y∂y,Sxfx,y=∫0xft,ytdt,Syfx,y=∫0yfx,ttdt,Jfx,y=fx,x,Qαfx,y=xαfx,y.Thus, 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], M-polynomials and topological indices of V-phenylenic nanotubes and nanotori were evaluated. For some current works on M-polynomials, readers are referred to [30–32]. 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]. Three decades ago, a strong association between commutative algebra and graph theory was established by means of a zero divisor graph. Graphs associated to finite commutative rings is a fast-developing area and are seen to have important applications in numerous areas like algebraic cryptography and coding theory, information and communication theory, and so on [36–39]. The input model of a frequency assignment network is a zero-divisor graph of commutative ring [40]. Let ZR and UR 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 defined a graph taking the elements of R as vertices and two elements u,v∈R are connected iff uv=0. Then Anderson and Livington [42] introduced the zero divisor graph ΓR, where the nonzero zero-divisors are taken as vertices and two vertices u,v are adjacent iff uv=0. In [43], Anderson and Badawi introduced total graph of a commutative ring with unity. The total graph of R, denoted by TR, is formed by considering the elements of R as vertices and two vertices u,v∈VTR are connected by an edge iff u+v∈ZR. 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. Thus, the derivation of such invariants for structures defined on a finite commutative ring is beneficial 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 ℤn, 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 [46–48]. The goal of the present work is to obtain different distance-based, degree-distance-based, and degree-based topological indices of the total graph Tℤnn∈ℤ+, the zero divisor graph Γℤrn (r is prime, n∈ℤ+), and the zero divisor graph Γℤr×ℤs×ℤt (r,s,t are primes) using Hosoya polynomial, Scultz polynomial, modified Scultz polynomial, and M-polynomial.


## 2. Motivation

The journey of topological indices was started through the Wiener index [1] in 1947, where the boiling points of the paraffins were modelled as follows:(26)tB=aw+bp+c,where tB 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. The quantitative structure-property relationships were established between boiling points and hyper-Wiener index in a series of cyclic and acyclic alkanes [49]. The first and second Zagreb indices were shown to be effective in the computation of the total π-electron energy of molecule [50]. They were suggested for the approximation of stretched carbon-skeleton [13]. The linear combination of the forgotten topological index and the first 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. The 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 identified. The study of graphs built from commutative rings focuses on the interaction between the algebraic and graph theoretical properties of the associated graph. This 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.


## 3. Methodology

Our main findings 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 graphTℤnn∈ℤ+, the zero divisor graph Γℤrn (r is prime, n∈ℤ+), and the zero divisor graph Γℤr×ℤs×ℤt (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, modified Schultz polynomial, and the M-polynomial. Using these polynomials, we derive degree based, degree-distance based, and distance based indices.


## 4. Main Results

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


### 4.1. Total GraphTℤn

Throughout this section, the ring of integers modulonn∈ℤ+ is considered, i.e., R=ℤn. From the fundamental theorem of arithmetic, we can express n=∏i=1kpiαi, where p1<p2<…<pk. Let the number of zero divisors of ℤn be m. We start with the following lemma.Observation 1.
(see [51]). The degree of any vertex v∈VTℤn is as follows:(i)
Ifn is odd, then we have(27)dv=m−1,v∈Zℤn,m,v∈Uℤn.(ii)
Ifn is even, then we have dv=m−1.Observation 2.
(see [44]). The cardinality of the edge set ETℤn is as follows:(28)ETℤn=n−12,nis odd prime,nm−12,nis even,mn−12,nis odd but not prime.Now, we compute theM-polynomial of the total graph ΓZn in Theorem 1.Theorem 1.
LetG be the total graph TZn. Then, we have(29)MG=nm−12xm−1ym−1,n≠2α1is even,E1xm−1ym−1+E2xm−1ym+E3xmym,n≠p1α1is odd.Proof.
The edge set ofTZn can be partitioned as follows:(30)ETZn=E1∪E2∪E3,where(31)E1=uv∈ETZn:u,v∈Zℤn,E2=uv∈ETZn:u∈Zℤn−0,v∈Uℤn,E3=uv∈ETZn:u,v∈Uℤn.
Also,E1=∑i=1kCn/pi2, E2=∑i=1km−n/pin/pi−1, E3=mn−1/2−∑i=1kCn/pi2+∑i=1km−n/pin/pi−1. From the definition, the M-polynomial of G is obtained as follows:(32)MG=∑i⩽jMi,jxiyj.
Now,(33)MG=Mm−1,m−1xm−1ym−1,n≠2α1is even,Mm−1,m−1xm−1ym−1+Mm−1,mxm−1ym+Mm,mxmym,n≠p1α1is odd.
Whenn≠2α1 is even, Mm−1,m−1=nm−1/2. When n≠p1α1 is odd, Mm−1,m−1=E1, Mm−1,m=E2, and Mm,m=E3. Thus, we have(34)MG=nm−12xm−1ym−1,n≠2α1is even,E1xm−1ym−1+E2xm−1ym+E3xmym,n≠p1α1is odd.
This completes the proof.
Now, using thisM-polynomial, we calculate some degree based topological index of the total graph Tℤn in the following theorem.Theorem 2.
LetG be the total graph Tℤn. Then, we have(1)
M1G=nm−12,n≠2α1is even,2m−1E1+2m−1E2+2mE3,n≠p1α1is odd.(2)
M2G=nm−13/2,n≠2α1is even,m−12E1+mm−1E2+m2E3,n≠p1α1is odd.(3)
FG=nm−13,n≠2α1is even,2m−12E1+2m2−2m+1E2+2m2E3,n≠p1α1is odd.(4)
M2mG=n/2m−1,n≠2α1is even,E1/m−12+E2/mm−1+E3/m2,n≠p1α1is odd.(5)
RαG=nm−12α+1/2,n≠2α1is even,m−12αE1+m2−mαE2+m2αE3,n≠p1α1is odd.(6)
RRαG=n/2m−12α−1,n≠2α1is even,E1/m−12α+E2/m2−mα+E3/m2α,n≠p1α1is odd.(7)
SDDG=nm−1,n≠2α1is even,2E1+2m2−2m+1/mm−1E2+2E3,n≠p1α1is odd.(8)
HG=n/2,n≠2α1is even,E1/m−1+2E2/2m−1+E3/m,n≠p1α1is odd.(9)
IG=nm−12/4,n≠2α1is even,m−1E1/2+m2−mE2/2+mE3/2,n≠p1α1is odd.(10)
AG=nm−17/16m−23,n≠2α1is even,m−16E1/8m−23+m2−m3E2/22m−33+m6E3/8m−13,n≠p1α1is odd..Proof.
Let(35)MG=fx,y=nm−12xm−1ym−1,n≠2α1is even,E1xm−1ym−1+E2xm−1ym+E3xmym,n≠p1α1is odd.
Then, we have(1)
Dx+Dyfx,y=nm−12xm−1ym−1,n≠2α1is even,2m−1E1xm−1ym−1+2m−1E2xm−1ym+2mE3xmym,n≠p1α1is odd.(2)
DxDyfx,y=nm−13/2xm−1ym−1,n≠2α1is even,m−12E1xm−1ym−1+mm−1E2xm−1ym+m2E3xmym,n≠p1α1is odd.(3)
Dx2+Dy2fx,y=nm−13xm−1ym−1,n≠2α1is even,2m−12E1xm−1ym−1+2m2−2m+1E2xm−1ym+2m2E3xmym,n≠p1α1is odd.(4)
SxSyfx,y=n/2m−1xm−1ym−1,n≠2α1is even,E1/m−12xm−1ym−1+E2/mm−1xm−1ym+E3/m2xmym,n≠p1α1is odd.(5)
DxαDyαfx,y=nm−12α+1/2xm−1ym−1,n≠2α1is even,m−12αE1xm−1ym−1+m2−mαE2xm−1ym+m2αE3xmym,n≠p1α1is odd.(6)
SxαSyαfx,y=n/2m−12α−1xm−1ym−1,n≠2α1is even,E1/m−12αxm−1ym−1+E2/m2−mαxm−1ym+E3/m2αxmym,n≠p1α1is odd.(7)
DxSy+SxDyfx,y=nm−12xm−1ym−1,n≠2α1is even,2E1xm−1ym−1+2m2−2m+1/mm−1E2xm−1ym+2E3xmym,n≠p1α1is odd.(8)
SxJfx,y=n/4x2m−1,n≠2α1is even,E1/2m−1x2m−1+E2/2m−1x2m−1+E3/2mx2m,n≠p1α1is odd.(9)
SxJDxDyfx,y=nm−12/4x2m−1,n≠2α1is even,m−1E1/2x2m−1+m2−mE2/2m−1x2m−1+mE3/2x2m,n≠p1α1is odd.(10)
Sx3Q−2JDx3Dy3fx,y=nm−17/16m−23x2m−2,n≠2α1is even,m−16E1/8m−23x2m−2+m2−m3E2/2m−33x2m−3+m6E3/8m−13x2m−1,n≠p1α1is odd..
Using Table1, we can easily obtain the required result.Table 2
Topological indices for the total graphTℤn, where n=6,10,15.
nM1ΓℤnM2ΓℤnFΓℤnM2mΓℤnRαΓℤnRRαΓℤnSDDΓℤnHΓℤnIΓℤnAΓℤn65481162132α+1321−α18313.5102.516102506251250152α+1521−α50562.5762.93915644212042561.151362α+1642α+2049α136−2α+1642−α+2049−α98.3817.485160.6922858.82Example 1.
Using Theorem2, we obtain the degree based topological indices of the total graph Tℤn for n=6,10,15 in Table 2.


### 4.2. Zero Divisor GraphΓℤrn

In this section, we obtain some distance- and degree-distance-based indices of the zero divisor graphΓℤrn, where r is prime and n is natural number.Theorem 3.
LetG be the zero divisor graph Γℤrn. Then, the Hosoya polynomial of G is given by(36)HG,x=rn−1−1+I1x+I2x2,nis even,rn−1−1+J1x+J2x2,nis odd,where I1=rn/2−1rn/2−2/2+r−1n/2−1rn−1−rn/2rn/2−1−1/r−1, I2=rn−1−rn/2rn−1−rn/2−1/2+r−1rnrn/2−1−1/r−1−n/2−1rn−1, J1=rn−1/2−1rn−1/2−2/2+r−1n−1/2rn−1−rn−1/2rn−1/2−1/r−1, and J2=rn−1−rn−1/2rn−1−rn−1/2−1/2+r−1rnrn−3/2−1/r−1−n−3/2rn−1.Proof.
The collection of all non-zero zero divisors ofℤrn is as follows:(37)Z∗ℤrn=r,2r,3r,…,rn−1−1r.
It is clear thatZ∗ℤrn=rn−1−1. We can partition the set Z∗ℤrn as follows:(38)Z∗ℤrn=P1∪P2∪…∪Pn−1,where(39)P1=k1r:k1=1,2,…,rn−1−1andr∤k1,P2=k2r2:k2=1,2,…,rn−2−1andr∤k2,⋮Pi=kiri:ki=1,2,…,rn−i−1andr∤ki,⋮Pn−1=kn−1rn−1:kn−1=1,2,…,r−1andr∤kn−1.
The rest of the proof can be constructed by considering the following two cases.Case 1. n is even.All the nodes belong to the setsPn/2,Pn/2+1,…,Pn−1 are mutually connected. Total number of such nodes is rn/2−1. Thus total number of such edges is rn/2−1rn/2−2/2=I11d. Each vertex of Pi is adjacent to every vertex of Pn−1,Pn−2,…,Pn−i and we denote this adjacency by Pi−Pn−1,Pn−2,…,Pn−i, where i=1,2,…,n/2−1. We describe such connections in Table 3.LetI12 be the total number of edges described in Table 3. Then, we have(40)I12=r−1rn−1−rn−2+rn−1−rn−3+rn−1−rn−4+,…,+rn−1−rn/2=rn−1−rn/2rn−1−rn/2−12+r−1rnrn/2−1−1r−1−n2−1rn−1.Thus, the total number of edges ofΓℤrn is I11+I12=I1.No two nodes belong to the partitionsP1,P2,…,Pn/2−1 are connected. From P1 to Pn/2−1, each pair of nodes have distance 2. Total number of such pairs is rn−1−rn/2rn−1−rn/2−1/2=I21, say. If u∈Pi and v∈ any member of Pn/2,Pn/2+1,…,Pn−i−1, then du,v=2, where i=1,2,…,n/2−1. We denote such pair by Pi−Pn/2,Pn/2+1,…,Pn−i−1 describe in detail in Table 4.LetI22 be the total number of edges described in Table 4. Then, we have(41)I22=r−1r3n/2−2−rn−1+r3n/2−3−rn−1+,…,+rn−rn−1=r−1rnrn/2−1−1r−1−n2−1rn−1.Thus, total number of pairsu,v such that du,v=2 is I21+I22=I2. Thus, using equation (2), we obtain the Hosoya polynomial of G as follows:(42)HG,x=∑κ=02dG,κxκ=rn−1−1+I11+I12x+I21+I22x2=rn−1−1+I1x+I2x2.Case 2. n is odd.
All the nodes belong to the setsPn+1/2,Pn/2+1,…,Pn−1 are mutually connected. Total number of such nodes is rn−1/2−1. Thus total number of such edges is rn−1/2−1rn−1/2−2/2=J11, say. Each vertex of Pi is adjacent to each vertex of Pn−1,Pn−2,…,Pn−i, where i=1,2,…,n−1/2. Similar to the first case, we can say that the total number of edges of Γℤrn is J11+J12, where(43)J12=r−1rn−1−rn−2+rn−1−rn−3+,…,+rn−1−rn−1/2=r−1n−12rn−1−rn−1/2rn−1/2−1r−1.
It is clear thatJ11+J12=J1. No two nodes belong to the partitions P1,P2,…,Pn−1/2 are connected. From P1 to Pn−1/2, each pair of nodes have distance 2. Total number of such pairs is rn−1−rn−1/2rn−1−rn−1/2−1/2=J21. If u∈Pi and v∈ any member of Pn+1/2,Pn+1/2+1,…,Pn−i−1, then du,v=2, where i=1,2,…,n−3/2. Likewise Case 1, total number of pairs u,v with du,v=2 is J21+J22, where(44)J22=r−1r3n−1/2−2−rn−1+r3n−1/2−3−rn−1+,…,+rn−rn−1=r−1rnrn−3/2−1r−1−n−32rn−1.
It is obvious thatJ21+J22=J2. Thus, using equation (2), we obtain the Hosoya polynomial of G as follows:(45)HG,x=∑κ=02dG,κxκ=rn−1−1+J11+J12x+J21+J22x2=rn−1−1+J1x+J2x2.
Using equation (3) and Theorem 3, we obtain the Wiener index of Γℤrn in Theorem 4.Table 3
Details of edge connections.
Edge connectionNumber of edgesP1−Pn−1r−1rn−1−rn−2P2−Pn−1,Pn−2r2−1rn−2−rn−3⋮⋮Pi−Pn−1,Pn−2,…,Pn−iri−1rn−i−rn−i−1⋮⋮Pn/2−1−Pn−1,Pn−2,…,Pn/2+1rn/2−1−1rn/2+1−rn/2Table 4
Details of pairs having distance 2.
PairsNumber of pairsP1−Pn/2,Pn/2+1,…,Pn−2rn−1−rn−2rn/2−rP2−Pn/2,Pn/2+1,…,Pn−3rn−2−rn−3rn/2−rr2⋮⋮Pi−Pn/2,Pn/2+1,…,Pn−i−1rn−i−rn−i−1rn/2−ri⋮⋮Pn/2−1−Pn/2rn/2+1−rn/2rn/2−rn/2−1Theorem 4.
LetG be the zero divisor graph Γℤrn. Then, the Wiener index of G is given by(46)WG=I1+2I2,nis even,J1+2J2,nis odd,where I1, I2, J1, and J2 are given in Theorem 3.Using equation (5) and Theorem 3, we obtain the hyper-Wiener index of Γℤrn in Theorem 5.Theorem 5.
LetG be the zero divisor graph Γℤrn. Then, the hyper-Wiener index of G is given by(47)WWG=I1+3I2,nis even,J1+3J2,nis odd,where I1, I2, J1, and J2 are given in Theorem 3.Theorem 6.
LetG be the zero divisor graph Γℤrn. Then, the Schultz polynomial of G is given by(48)ScG,x=L1+L2+L3x+L1′+L2′+L3′x2,nis even,Q1+Q2+Q3x+Q1′+Q2′+Q3′x2,nis odd,where Li, Li′, Qi, and Qi′, i=1,2,3, are described in proof.Proof.
To construct the proof, we consider the partition ofΓℤrn as illustrated in the proof of Theorem 3. The degrees of vertices G are as follows:(49)dui=ri−1,i<n2,nis even,ri−2,i≥n2,nis even,ri−1,i<n+12,nis odd,ri−2,i≥n+12,nis odd,where ui∈Pi, i=1,2,…,n−1. We consider the following two cases.Case 1. n is even.First, we consider the partitions of all pairsu,v having distance 1. We know that du,v=1 for all u,v∈Pi, i=n/2,n/2+1,…,n−1. We describe the partitions of all such pairs u,v in Table 5.LetL1=∑k=0n/2−1Fkrn/2+k−2+rn/2+k−2=∑k=0n/2−1Fk2rn/2+k−4. If u∈Pi, v∈Pj, i,j=n/2,n/2+1,…,n−1, i≠j, then du,v=1. We describe the partitions of all such pairs u,v in Table 6.LetL2=1/2∑k=0n/2−1∑l=0n/2−1Fklrn/2+k+rn/2+l−4, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n/2−1,j=n/2+1,n/2+2,…,n−1, i+j≥n, then du,v=1. We describe the partitions of all such pairs u,v in Table 7.LetL3=∑k=1n/2−1∑l=1kFkl′rk+rn−l−3.Now, we consider the partitions of all pairsu,v having distance 2. We have du,v=2 for all u,v∈Pi, i=1,2,…,n/2−1. We describe the partitions of all such pairs u,v in Table 8.LetL1′=∑k=1n/2−1fkrk−1+rk−1=∑k=1n/2−1fk2rk−2. If u∈Pi, v∈Pj, i,j=1,2,…,n/2−1, i≠j, then du,v=2. We describe the partitions of all such pairs u,v in Table 9.LetL2′=1/2∑k=1n/2−1∑l=1n/2−1fklrk+rl−2, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n/2−1,j=n/2,n/2+1,…,n−2, i+j<n, then du,v=2. We describe the partitions of all such pairs u,v in Table 10.LetL3′=∑k=1n/2−1∑l=0n/2−1fkl′rn/2−k+rn/2+l−3.Case 2. n is odd.
First, we consider the partitions of all pairsu,v with du,v=1. We know that du,v=1 for all u,v∈Pi, i=n+1/2,n+1/2+1,…,n−1. We describe the partitions of all such pairs u,v in Table 11.
LetQ1=∑k=0n−3/2Ck2rn+1/2+k−4. If u∈Pi, v∈Pj, i,j=n+1/2,n+1/2+1,…,n−1, i≠j, then du,v=1. We describe the partitions of all such pairs u,v in Table 12.
LetQ2=1/2∑k=0n−3/2∑l=0n−3/2Cklrn+1/2+k+rn+1/2+l−4, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n−1/2,j=n+1/2,n+3/2,…,n−1, i+j≥n, then du,v=1. We describe the partitions of all such pairs u,v in Table 13.
LetQ3=∑k=1n−1/2∑l=1kCkl′rk+rn−l−3.
Now, we consider the partitions of all pairsu,v having distance 2. We have du,v=2 for all u,v∈Pi, i=1,2,…,n−1/2. We describe the partitions of all such pairs u,v in Table 14.
LetQ1′=∑k=1n−1/2ck2rk−2. If u∈Pi, v∈Pj, i,j=1,2,…,n−1/2, i≠j, then du,v=2. We describe the partitions of all such pairs u,v in Table 15.
LetQ2′=1/2∑k=1n−1/2∑l=1n−1/2cklrk+rl−2, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n−3/2,j=n+1/2,n+3/2,…,n−2, i+j<n, then du,v=2. We describe the partitions of all such pairs u,v in Table 16.
LetQ3′=∑k=0n−5/2∑l=0kckl′rn−3/2−k+rn+1/2+l−3.
Using equation (10) and Theorem 6, we obtain the Schultz index of Γℤrn in Theorem 7.Table 5
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn/2+k−2,rn/2+k−2C2rn/2−k−rn/2−k−1=Fk, say, where k=0,1,2,…,n/2−1Table 6
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn/2+k−2,rn/2+l−2rn/2−k−rn/2−k−1rn/2−l−rn/2−l−1=Fkl, say, where k,l=0,1,2,…,n/2−1, k≠lTable 7
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrk−1,rn−l−2rn−k−rn−k−1rl−rl−1=Fkl′, say, where k,l satisfy the following relation: k=1,2,…,n/2−1 and for each k, l=1,2,…,kTable 8
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rk−1C2rn−k−rn−k−1=fk, say, where k=1,2,…,n/2−1Table 9
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rl−1rn−k−rn−k−1rn−l−rn−l−1=fkl, say, where k,l=1,2,…,n/2−1, k≠lTable 10
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrn/2−k−1,rn/2+l−2rn/2+k−rn/2+k−1rn/2−l−rn/2−l−1=fkl′, say, where k,l satisfy the following relation: k=1,2,…,n/2−1 and for each k, l=0,1,…,k−1Table 11
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn+1/2+k−2,rn+1/2+k−2C2rn−1/2−k−rn−3/2−k=Ck, say, where k=0,1,2,…,n−3/2Table 12
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn+1/2+k−2,rn+1/2+l−2rn−1/2−k−rn−3/2−krn−1/2−l−rn−3/2−k=Ckl, say, where k,l=0,1,2,…,n−3/2, k≠lTable 13
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrk−1,rn−l−2rn−k−rn−k−1rl−rl−1=Ckl′, say, where k,l satisfy the following relation: k=1,2,…,n−1/2 and for each k, l=1,2,…,kTable 14
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rk−1C2rn−k−rn−k−1=ck, say, where k=1,2,…,n−1/2Table 15
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rl−1rn−k−rn−k−1rn−l−rn−l−1=ckl, say, where k,l=1,2,…,n−1/2, k≠lTable 16
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrn−3/2−k−1,rn+1/2+l−2rn+3/2+k−rn+1/2+krn−1/2−l−rn−1/2−l−1=ckl′, say, where k,l satisfy the following relation: k=0,1,…,n−5/2 and for each k, l=0,1,…,kTheorem 7.
LetG be the zero divisor graph Γℤrn. Then, the Schultz index of G is given by(50)ScG=L1+L2+L3+2L1′+L2′+L3′,nis even,Q1+Q2+Q3+2Q1′+Q2′+Q3′,nis odd,where Li, Li′, Qi, Qi′, i=1,2,3, are described in proof of Theorem 6.Theorem 8.
LetG be the zero divisor graph Γℤrn. Then, the modified Schultz polynomial of G is given by(51)Sc∗G,x=L1+L2+L3x+L1′+L2′+L3′x2,nis even,Q1+Q2+Q3x+Q1′+Q2′+Q3′x2,nis odd,where L1=∑k=0n/2−1Fkrn/2+k−22, L2=1/2∑k=0n/2−1∑l=0n/2−1Fklrn/2+k−2rn/2+l−2, L3=∑k=1n/2−1∑l=1kFkl′rk−1rn−l−2, L1′=∑k=1n/2−1fkrk−12, L2′=1/2∑k=1n/2−1∑l=1n/2−1fklrk−1rl−1, L3′=∑k=1n/2−1∑l=0k−1fkl′rn/2−k−1rn/2+l−2, Q1=∑k=0n−3/2−1Ckrn+1/2+k−22, Q2=1/2∑k=0n−3/2∑l=0n−3/2Cklrn+1/2+k−2rn+1/2+l−2, Q3=∑k=1n−1/2∑l=1kCkl′rk−1rn−l−2, Q1′=∑k=1n−1/2ckrk−12, Q2′=1/2∑k=1n−1/2∑l=1n−1/2cklrk−1rl−1, and Q3′=∑k=0n−5/2∑l=0kckl′rn−3/2−k−1rn+1/2+l−2. Fk,Fkl,Fkl′,fk,fkl,fkl′,Ck,Ckl,Ckl′,ck,ckl, and ckl′ are illustrated in the proof of Theorem 6.Using equation (11) and Theorem 8, we obtain the modified Schultz index of Γℤrn in Theorem 9.Theorem 9.
LetG be the zero divisor graph Γℤrn. Then, the modified Schultz index of G is given by(52)Sc∗G=L1+L2+L3+2L1′+L2′+L3′,nis even,Q1+Q2+Q3+2Q1′+Q2′+Q3′,nis odd,where Li, Li′, Qi, and Qi′, i=1,2,3, are described in Theorem 8.


### 4.3. Zero Divisor GraphΓℤr×ℤs×ℤt

In this section, we obtain some distance-based, degree-distance-based, and degree-based indices of the zero divisor graphΓℤr×ℤs×ℤt, where r,s,t are primes. The structure of the zero divisor graph Γℤ5×ℤ3×ℤ2 is shown in Figure 1.Figure 1
The zero divisor graphΓℤ5×ℤ3×ℤ2.We start by computing the Hosoya polynomial in Theorem10.Theorem 10.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, its Hosoya polynomial is given by(53)HG,x=A0+A1x+A2x2+A3x3,where Aα is the number of pair of nodes having distance α.Proof.
The number of nodes and edges ofG are rs+st+rs−r−s−t and r−1s−1+r−1t−1+s−1t−1+3r−1s−1t−1. Thus, A0=rs+st+rs−r−s−t and A1=r−1s−1+r−1t−1+s−1t−1+3r−1s−1t−1. Now, we find A2 and A3. We list the information about all pairs of nodes of G having distance 2 in Table 17.
Thus, we haveA2 = 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.
Thus, we haveA3 = 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 Γℤr×ℤs×ℤt in Theorem 11.Table 17
Information related to the pair of nodes ofG having distance 2.
x,y,x,y∈VGNumber of pairsRemarks0,0,w1,0,0,w2C2t−10<w1,w2<t0,0,w1,0,vj,w2s−1t−120<w1,w2<t, 0<vj<s0,0,w1,ui,0,w2r−1t−120<w1,w2<t, 0<ui<r0,v1,0,0,v2,0C2s−10<v1,v2<s0,v1,0,ui,v2,0r−1s−120<ui<r, 0<v1,v2<su1,0,0,u2,0,0C2r−10<u1,u2<ru1,0,0,u2,vj,0r−12s−10<u1,u2<r, 0<vj<su1,0,0,u2,0,wkr−12t−10<u1,u2<r, 0<wk<tu1,v1,0,u2,v2,0C2r−1s−10<u1,u2<r, 0<v1,v2<s0,v1,w1,0,v2,w2C2s−1t−10<v1,v2<s, 0<w1,w2<tu1,0,w1,u2,0,w2C2r−1t−10<u1,u2<r, 0<w1,w2<tTable 18
Information related to the pair of nodes ofG having distance 3.
x,y,x,y∈VGNumber of pairsRemarksu1,vj,0,u2,0,wkr−12s−1t−10<u1,u2<r, 0<vj<s, 0<wk<tui,v1,0,0,v2,wkr−1s−12t−10<ui<r, 0<v1,v2<s, 0<wk<t0,vj,w1,ui,0,w2r−1s−1t−120<ui<r, 0<vj<s, 0<w1,w2<tTheorem 11.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, we have(54)WG=A1+2A2+3A3,WWG=A1+3A2+6A3,where Aα 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 Lemma1.Lemma 1.
The degrees of the vertices of the graphΓℤr×ℤs×ℤt are given by(i)
du,0,0=st−1,0<u<r(ii)
d0,v,0=rt−1,0<v<s(iii)
d0,0,w=rs−1,0<w<t(iv)
du,v,0=t−1,0<u<r,0<v<s(v)
d0,v,w=r−1,0<v<s,0<w<t(vi)
du,0,w=s−1,0<u<r,0<w<tNow, we compute theM-polynomial of the zero divisor graph Γℤr×ℤs×ℤt in the following theorem.Theorem 12.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, we have(55)MG=r−1s−1xst−1yrt−1+r−1t−1xst−1yrs−1+s−1t−1xrt−1yrs−1+r−1s−1t−1xr−1yst−1+xs−1yrt−1+xt−1yrs−1.Proof.
LetG be the total graph Γℤr×ℤs×ℤt. The edge set of G can be partitioned as follows:(56)EG=E1∪E2∪E3∪E4∪E5∪E6,where(57)E1=u,0,00,v,0∈EG:0<u<r,0<v<s,E2=u,0,00,0,w∈EG:0<u<r,0<w<t,E3=0,v,00,0,w∈EG:0<v<s,0<w<t,E4=u,0,00,v,w∈EG:0<u<r,0<v<s,0<w<t,E5=0,v,0u,0,w∈EG:0<u<r,0<v<s,0<w<t,E6=0,0,wu,v,0∈EG:0<u<r,0<v<s,0<w<t.
Also,E1=r−1s−1, E2=r−1t−1, E3=s−1t−1, and E4=E5=E6=r−1s−1t−1. From the definition, the M-polynomial of G is obtained as follows:(58)MG=∑i⩽jMi,jxiyj.
Now,(59)MG=Mst−1,rt−1xst−1yrt−1+Mst−1,rs−1xst−1yrs−1+Mrt−1,rs−1xrt−1yrs−1+Mr−1,st−1xr−1yst−1+Ms−1,rt−1xs−1yrt−1+Mt−1,rs−1xt−1yrs−1=E1xst−1yrt−1+E2xst−1yrs−1+E3xrt−1yrs−1+E4xr−1yst−1+E5xs−1yrt−1+E6xt−1yrs−1.
After putting the values ofEi, i=1,2,…,6, we obtain the required result. This completes the proof.
Now, using thisM-polynomial, we calculate some degree-based topological indices of the zero divisor graph Γℤr×ℤs×ℤt in Theorem 13 like Theorem 2.Theorem 13.
LetG be the total graph Γℤr×ℤs×ℤt. Then, we have(1)
M1G=r−1s−1rt+st−2+r−1t−1rs+st−2+s−1t−1rs+rt−2+r−1s−1t−1rs+rt+st+r+s+t−6(2)
M2G=r−1s−1rt−1st−1+r−1t−1rs−1st−1+s−1t−1rs−1rt−1+r−1s−1t−1r−1st−1+s−1rt−1+t−1rs−1(3)
FG=r−1s−1rt−12+st−12+r−1t−1rs−12+st−12+s−1t−1rs−12+rt−12+r−1s−1t−1r−12+s−12+t−12+rs−12+st−12+rt−12(4)
M2mG=r−1s−1/rt−1st−1+r−1t−1/rs−1st−1+s−1t−1/rs−1rt−1+r−1s−1t−11/r−1st−1+1/s−1rt−1+1/t−1rs−1(5)
RαG=r−1s−1rt−1st−1α+r−1t−1rs−1st−1α+s−1t−1rs−1rt−1α+r−1s−1t−1r−1st−1α+s−1rt−1α+t−1rs−1α(6)
RRαG=r−1s−1rt−1st−1−α+r−1t−1rs−1st−1−α+s−1t−1rs−1rt−1−α+r−1s−1t−1r−1st−1−α+s−1rt−1−α+t−1rs−1−α(7)
SDDG=r−1s−1rt−1/st−1+rt−1/st−1+r−1t−1rs−1/st−1+st−1/rs−1+s−1t−1rs−1/rt−1+rt−1/rs−1+r−1s−1t−1r−1/st−1+st−1/r−1+s−1/rt−1+rt−1/s−1+t−1/rs−1+rs−1/t−1(8)
HG=2r−1s−1/rt+st−2+2r−1t−1/rs+st−2+2s−1t−1/rs+rt−2+2r−1s−1t−11/st+r−2+1/rt+s−2+1/rs+t−2(9)
IG=r−1s−1rt−1st−1/rt+st−2+r−1t−1rs−1st−1/rs+st−2+s−1t−1rs−1rt−1/rs+rt−2+r−1s−1t−1r−1st−1/r+st−2+s−1rt−1/s+rt−2+t−1rs−1/t+rs−2(10)
AG=r−1s−1rt−13st−13/rt+st−43+r−1t−1rs−13st−13/rs+st−43+s−1t−1rs−13rt−13/rs+rt−43+r−1s−1t−1r−13st−13/r+st−43+s−13rt−13/s+rt−43+t−13rs−13/t+rs−43.Now, we obtain the Schultz polynomial of the zero divisor graphΓℤr×ℤs×ℤt.Theorem 14.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, the Schultz and modified Schultz polynomials of G are given by(60)ScG,x=S1x+S2x2+S3x3,Sc∗G,x=S1∗x+S2∗x2+S3∗x3,where Si and Si∗, i=1,2,3, are described in proof.Proof.
From the definition of Scultz polynomial, we have(61)ScG,x=∑u,v⊆VGdu+dvxdu,v=∑du,v=1du+dvx+∑du,v=2du+dvx2+∑du,v=3du+dvx3.
Let∑du,v=idu+dv=Si, i=1,2,3. We have from Theorem 12, S1=r-1s-1rt+st-2+s-1t-1rt+rs-2+r-1s-1rt+st-2+r-1s-1t-1rs+st+rt+r+s+t-6. Now, using Table 17 and Lemma 1, we get S2=C2t-12rs-2+s-1t-12rs+r-2+r-1t-12rs+s-2+C2s-12rt-2+r-1s-12rt+t-2+C2r-12st-2+r-12s-1st+t-2+r-12t-1st+s-2+C2r-1s-12t-2+C2s-1t-12r-2+C2r-1t-12s-2. From Table 18 and Lemma 1, we get S3=r−12s−1t−1s+t−2+r−1s−12t−1r+t−2+r−1s−1t−12r+s−2. Putting the values of Si, i=1,2,3 in equation (61), we obtain the required result. From the definition of Scultz polynomial, we have(62)Sc∗G,x=∑u,v⊆VGdudvxdu,v=∑du,v=1dudvx+∑du,v=2dudvx2+∑du,v=3dudvx3.
Let∑du,v=idudv=Si∗, i=1,2,3. We have from Theorem 12, S1*=r-1s-1rt-1st-1+s-1t-1rs-1st-1+r-1s-1rt-1rs-1+r-1s-1t-1st-1r-1+rt-1s-1+rs-1t-1. Now using Table 17 and Lemma 1, we get S2*=C2t-1rs-12+s-1t-12rs-1r-1+r-1t-12rs-1s-1+C2s-1rt-12+r-1s-12rt-1t-1+C2r-1st-12+r-12s-1st-1t-1+r-12t-1st-1s-1+C2r-1s-1t-12+C2s-1t-1r-12+C2r-1t-1s-12. From Table 18 and Lemma 1, we get S3∗=3r−12s−12t−12. Putting the values of Si, i=1,2,3 in equation (62), we obtain the required result.
Now, using equations (10) and (11), we obtain the Schultz and modified Schultz indices in Theorem 15.Theorem 15.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, the Schultz and modified Schultz indices of G are given by(63)ScG=S1+2S2+3S3,Sc∗G=S1∗+2S2∗+3S3∗,where Si and Si∗, i=1,2,3, are described in proof of Theorem 14.


## 4.1. Total GraphTℤn

Throughout this section, the ring of integers modulonn∈ℤ+ is considered, i.e., R=ℤn. From the fundamental theorem of arithmetic, we can express n=∏i=1kpiαi, where p1<p2<…<pk. Let the number of zero divisors of ℤn be m. We start with the following lemma.Observation 1.
(see [51]). The degree of any vertex v∈VTℤn is as follows:(i)
Ifn is odd, then we have(27)dv=m−1,v∈Zℤn,m,v∈Uℤn.(ii)
Ifn is even, then we have dv=m−1.Observation 2.
(see [44]). The cardinality of the edge set ETℤn is as follows:(28)ETℤn=n−12,nis odd prime,nm−12,nis even,mn−12,nis odd but not prime.Now, we compute theM-polynomial of the total graph ΓZn in Theorem 1.Theorem 1.
LetG be the total graph TZn. Then, we have(29)MG=nm−12xm−1ym−1,n≠2α1is even,E1xm−1ym−1+E2xm−1ym+E3xmym,n≠p1α1is odd.Proof.
The edge set ofTZn can be partitioned as follows:(30)ETZn=E1∪E2∪E3,where(31)E1=uv∈ETZn:u,v∈Zℤn,E2=uv∈ETZn:u∈Zℤn−0,v∈Uℤn,E3=uv∈ETZn:u,v∈Uℤn.
Also,E1=∑i=1kCn/pi2, E2=∑i=1km−n/pin/pi−1, E3=mn−1/2−∑i=1kCn/pi2+∑i=1km−n/pin/pi−1. From the definition, the M-polynomial of G is obtained as follows:(32)MG=∑i⩽jMi,jxiyj.
Now,(33)MG=Mm−1,m−1xm−1ym−1,n≠2α1is even,Mm−1,m−1xm−1ym−1+Mm−1,mxm−1ym+Mm,mxmym,n≠p1α1is odd.
Whenn≠2α1 is even, Mm−1,m−1=nm−1/2. When n≠p1α1 is odd, Mm−1,m−1=E1, Mm−1,m=E2, and Mm,m=E3. Thus, we have(34)MG=nm−12xm−1ym−1,n≠2α1is even,E1xm−1ym−1+E2xm−1ym+E3xmym,n≠p1α1is odd.
This completes the proof.
Now, using thisM-polynomial, we calculate some degree based topological index of the total graph Tℤn in the following theorem.Theorem 2.
LetG be the total graph Tℤn. Then, we have(1)
M1G=nm−12,n≠2α1is even,2m−1E1+2m−1E2+2mE3,n≠p1α1is odd.(2)
M2G=nm−13/2,n≠2α1is even,m−12E1+mm−1E2+m2E3,n≠p1α1is odd.(3)
FG=nm−13,n≠2α1is even,2m−12E1+2m2−2m+1E2+2m2E3,n≠p1α1is odd.(4)
M2mG=n/2m−1,n≠2α1is even,E1/m−12+E2/mm−1+E3/m2,n≠p1α1is odd.(5)
RαG=nm−12α+1/2,n≠2α1is even,m−12αE1+m2−mαE2+m2αE3,n≠p1α1is odd.(6)
RRαG=n/2m−12α−1,n≠2α1is even,E1/m−12α+E2/m2−mα+E3/m2α,n≠p1α1is odd.(7)
SDDG=nm−1,n≠2α1is even,2E1+2m2−2m+1/mm−1E2+2E3,n≠p1α1is odd.(8)
HG=n/2,n≠2α1is even,E1/m−1+2E2/2m−1+E3/m,n≠p1α1is odd.(9)
IG=nm−12/4,n≠2α1is even,m−1E1/2+m2−mE2/2+mE3/2,n≠p1α1is odd.(10)
AG=nm−17/16m−23,n≠2α1is even,m−16E1/8m−23+m2−m3E2/22m−33+m6E3/8m−13,n≠p1α1is odd..Proof.
Let(35)MG=fx,y=nm−12xm−1ym−1,n≠2α1is even,E1xm−1ym−1+E2xm−1ym+E3xmym,n≠p1α1is odd.
Then, we have(1)
Dx+Dyfx,y=nm−12xm−1ym−1,n≠2α1is even,2m−1E1xm−1ym−1+2m−1E2xm−1ym+2mE3xmym,n≠p1α1is odd.(2)
DxDyfx,y=nm−13/2xm−1ym−1,n≠2α1is even,m−12E1xm−1ym−1+mm−1E2xm−1ym+m2E3xmym,n≠p1α1is odd.(3)
Dx2+Dy2fx,y=nm−13xm−1ym−1,n≠2α1is even,2m−12E1xm−1ym−1+2m2−2m+1E2xm−1ym+2m2E3xmym,n≠p1α1is odd.(4)
SxSyfx,y=n/2m−1xm−1ym−1,n≠2α1is even,E1/m−12xm−1ym−1+E2/mm−1xm−1ym+E3/m2xmym,n≠p1α1is odd.(5)
DxαDyαfx,y=nm−12α+1/2xm−1ym−1,n≠2α1is even,m−12αE1xm−1ym−1+m2−mαE2xm−1ym+m2αE3xmym,n≠p1α1is odd.(6)
SxαSyαfx,y=n/2m−12α−1xm−1ym−1,n≠2α1is even,E1/m−12αxm−1ym−1+E2/m2−mαxm−1ym+E3/m2αxmym,n≠p1α1is odd.(7)
DxSy+SxDyfx,y=nm−12xm−1ym−1,n≠2α1is even,2E1xm−1ym−1+2m2−2m+1/mm−1E2xm−1ym+2E3xmym,n≠p1α1is odd.(8)
SxJfx,y=n/4x2m−1,n≠2α1is even,E1/2m−1x2m−1+E2/2m−1x2m−1+E3/2mx2m,n≠p1α1is odd.(9)
SxJDxDyfx,y=nm−12/4x2m−1,n≠2α1is even,m−1E1/2x2m−1+m2−mE2/2m−1x2m−1+mE3/2x2m,n≠p1α1is odd.(10)
Sx3Q−2JDx3Dy3fx,y=nm−17/16m−23x2m−2,n≠2α1is even,m−16E1/8m−23x2m−2+m2−m3E2/2m−33x2m−3+m6E3/8m−13x2m−1,n≠p1α1is odd..
Using Table1, we can easily obtain the required result.Table 2
Topological indices for the total graphTℤn, where n=6,10,15.
nM1ΓℤnM2ΓℤnFΓℤnM2mΓℤnRαΓℤnRRαΓℤnSDDΓℤnHΓℤnIΓℤnAΓℤn65481162132α+1321−α18313.5102.516102506251250152α+1521−α50562.5762.93915644212042561.151362α+1642α+2049α136−2α+1642−α+2049−α98.3817.485160.6922858.82Example 1.
Using Theorem2, we obtain the degree based topological indices of the total graph Tℤn for n=6,10,15 in Table 2.


## 4.2. Zero Divisor GraphΓℤrn

In this section, we obtain some distance- and degree-distance-based indices of the zero divisor graphΓℤrn, where r is prime and n is natural number.Theorem 3.
LetG be the zero divisor graph Γℤrn. Then, the Hosoya polynomial of G is given by(36)HG,x=rn−1−1+I1x+I2x2,nis even,rn−1−1+J1x+J2x2,nis odd,where I1=rn/2−1rn/2−2/2+r−1n/2−1rn−1−rn/2rn/2−1−1/r−1, I2=rn−1−rn/2rn−1−rn/2−1/2+r−1rnrn/2−1−1/r−1−n/2−1rn−1, J1=rn−1/2−1rn−1/2−2/2+r−1n−1/2rn−1−rn−1/2rn−1/2−1/r−1, and J2=rn−1−rn−1/2rn−1−rn−1/2−1/2+r−1rnrn−3/2−1/r−1−n−3/2rn−1.Proof.
The collection of all non-zero zero divisors ofℤrn is as follows:(37)Z∗ℤrn=r,2r,3r,…,rn−1−1r.
It is clear thatZ∗ℤrn=rn−1−1. We can partition the set Z∗ℤrn as follows:(38)Z∗ℤrn=P1∪P2∪…∪Pn−1,where(39)P1=k1r:k1=1,2,…,rn−1−1andr∤k1,P2=k2r2:k2=1,2,…,rn−2−1andr∤k2,⋮Pi=kiri:ki=1,2,…,rn−i−1andr∤ki,⋮Pn−1=kn−1rn−1:kn−1=1,2,…,r−1andr∤kn−1.
The rest of the proof can be constructed by considering the following two cases.Case 1. n is even.All the nodes belong to the setsPn/2,Pn/2+1,…,Pn−1 are mutually connected. Total number of such nodes is rn/2−1. Thus total number of such edges is rn/2−1rn/2−2/2=I11d. Each vertex of Pi is adjacent to every vertex of Pn−1,Pn−2,…,Pn−i and we denote this adjacency by Pi−Pn−1,Pn−2,…,Pn−i, where i=1,2,…,n/2−1. We describe such connections in Table 3.LetI12 be the total number of edges described in Table 3. Then, we have(40)I12=r−1rn−1−rn−2+rn−1−rn−3+rn−1−rn−4+,…,+rn−1−rn/2=rn−1−rn/2rn−1−rn/2−12+r−1rnrn/2−1−1r−1−n2−1rn−1.Thus, the total number of edges ofΓℤrn is I11+I12=I1.No two nodes belong to the partitionsP1,P2,…,Pn/2−1 are connected. From P1 to Pn/2−1, each pair of nodes have distance 2. Total number of such pairs is rn−1−rn/2rn−1−rn/2−1/2=I21, say. If u∈Pi and v∈ any member of Pn/2,Pn/2+1,…,Pn−i−1, then du,v=2, where i=1,2,…,n/2−1. We denote such pair by Pi−Pn/2,Pn/2+1,…,Pn−i−1 describe in detail in Table 4.LetI22 be the total number of edges described in Table 4. Then, we have(41)I22=r−1r3n/2−2−rn−1+r3n/2−3−rn−1+,…,+rn−rn−1=r−1rnrn/2−1−1r−1−n2−1rn−1.Thus, total number of pairsu,v such that du,v=2 is I21+I22=I2. Thus, using equation (2), we obtain the Hosoya polynomial of G as follows:(42)HG,x=∑κ=02dG,κxκ=rn−1−1+I11+I12x+I21+I22x2=rn−1−1+I1x+I2x2.Case 2. n is odd.
All the nodes belong to the setsPn+1/2,Pn/2+1,…,Pn−1 are mutually connected. Total number of such nodes is rn−1/2−1. Thus total number of such edges is rn−1/2−1rn−1/2−2/2=J11, say. Each vertex of Pi is adjacent to each vertex of Pn−1,Pn−2,…,Pn−i, where i=1,2,…,n−1/2. Similar to the first case, we can say that the total number of edges of Γℤrn is J11+J12, where(43)J12=r−1rn−1−rn−2+rn−1−rn−3+,…,+rn−1−rn−1/2=r−1n−12rn−1−rn−1/2rn−1/2−1r−1.
It is clear thatJ11+J12=J1. No two nodes belong to the partitions P1,P2,…,Pn−1/2 are connected. From P1 to Pn−1/2, each pair of nodes have distance 2. Total number of such pairs is rn−1−rn−1/2rn−1−rn−1/2−1/2=J21. If u∈Pi and v∈ any member of Pn+1/2,Pn+1/2+1,…,Pn−i−1, then du,v=2, where i=1,2,…,n−3/2. Likewise Case 1, total number of pairs u,v with du,v=2 is J21+J22, where(44)J22=r−1r3n−1/2−2−rn−1+r3n−1/2−3−rn−1+,…,+rn−rn−1=r−1rnrn−3/2−1r−1−n−32rn−1.
It is obvious thatJ21+J22=J2. Thus, using equation (2), we obtain the Hosoya polynomial of G as follows:(45)HG,x=∑κ=02dG,κxκ=rn−1−1+J11+J12x+J21+J22x2=rn−1−1+J1x+J2x2.
Using equation (3) and Theorem 3, we obtain the Wiener index of Γℤrn in Theorem 4.Table 3
Details of edge connections.
Edge connectionNumber of edgesP1−Pn−1r−1rn−1−rn−2P2−Pn−1,Pn−2r2−1rn−2−rn−3⋮⋮Pi−Pn−1,Pn−2,…,Pn−iri−1rn−i−rn−i−1⋮⋮Pn/2−1−Pn−1,Pn−2,…,Pn/2+1rn/2−1−1rn/2+1−rn/2Table 4
Details of pairs having distance 2.
PairsNumber of pairsP1−Pn/2,Pn/2+1,…,Pn−2rn−1−rn−2rn/2−rP2−Pn/2,Pn/2+1,…,Pn−3rn−2−rn−3rn/2−rr2⋮⋮Pi−Pn/2,Pn/2+1,…,Pn−i−1rn−i−rn−i−1rn/2−ri⋮⋮Pn/2−1−Pn/2rn/2+1−rn/2rn/2−rn/2−1Theorem 4.
LetG be the zero divisor graph Γℤrn. Then, the Wiener index of G is given by(46)WG=I1+2I2,nis even,J1+2J2,nis odd,where I1, I2, J1, and J2 are given in Theorem 3.Using equation (5) and Theorem 3, we obtain the hyper-Wiener index of Γℤrn in Theorem 5.Theorem 5.
LetG be the zero divisor graph Γℤrn. Then, the hyper-Wiener index of G is given by(47)WWG=I1+3I2,nis even,J1+3J2,nis odd,where I1, I2, J1, and J2 are given in Theorem 3.Theorem 6.
LetG be the zero divisor graph Γℤrn. Then, the Schultz polynomial of G is given by(48)ScG,x=L1+L2+L3x+L1′+L2′+L3′x2,nis even,Q1+Q2+Q3x+Q1′+Q2′+Q3′x2,nis odd,where Li, Li′, Qi, and Qi′, i=1,2,3, are described in proof.Proof.
To construct the proof, we consider the partition ofΓℤrn as illustrated in the proof of Theorem 3. The degrees of vertices G are as follows:(49)dui=ri−1,i<n2,nis even,ri−2,i≥n2,nis even,ri−1,i<n+12,nis odd,ri−2,i≥n+12,nis odd,where ui∈Pi, i=1,2,…,n−1. We consider the following two cases.Case 1. n is even.First, we consider the partitions of all pairsu,v having distance 1. We know that du,v=1 for all u,v∈Pi, i=n/2,n/2+1,…,n−1. We describe the partitions of all such pairs u,v in Table 5.LetL1=∑k=0n/2−1Fkrn/2+k−2+rn/2+k−2=∑k=0n/2−1Fk2rn/2+k−4. If u∈Pi, v∈Pj, i,j=n/2,n/2+1,…,n−1, i≠j, then du,v=1. We describe the partitions of all such pairs u,v in Table 6.LetL2=1/2∑k=0n/2−1∑l=0n/2−1Fklrn/2+k+rn/2+l−4, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n/2−1,j=n/2+1,n/2+2,…,n−1, i+j≥n, then du,v=1. We describe the partitions of all such pairs u,v in Table 7.LetL3=∑k=1n/2−1∑l=1kFkl′rk+rn−l−3.Now, we consider the partitions of all pairsu,v having distance 2. We have du,v=2 for all u,v∈Pi, i=1,2,…,n/2−1. We describe the partitions of all such pairs u,v in Table 8.LetL1′=∑k=1n/2−1fkrk−1+rk−1=∑k=1n/2−1fk2rk−2. If u∈Pi, v∈Pj, i,j=1,2,…,n/2−1, i≠j, then du,v=2. We describe the partitions of all such pairs u,v in Table 9.LetL2′=1/2∑k=1n/2−1∑l=1n/2−1fklrk+rl−2, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n/2−1,j=n/2,n/2+1,…,n−2, i+j<n, then du,v=2. We describe the partitions of all such pairs u,v in Table 10.LetL3′=∑k=1n/2−1∑l=0n/2−1fkl′rn/2−k+rn/2+l−3.Case 2. n is odd.
First, we consider the partitions of all pairsu,v with du,v=1. We know that du,v=1 for all u,v∈Pi, i=n+1/2,n+1/2+1,…,n−1. We describe the partitions of all such pairs u,v in Table 11.
LetQ1=∑k=0n−3/2Ck2rn+1/2+k−4. If u∈Pi, v∈Pj, i,j=n+1/2,n+1/2+1,…,n−1, i≠j, then du,v=1. We describe the partitions of all such pairs u,v in Table 12.
LetQ2=1/2∑k=0n−3/2∑l=0n−3/2Cklrn+1/2+k+rn+1/2+l−4, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n−1/2,j=n+1/2,n+3/2,…,n−1, i+j≥n, then du,v=1. We describe the partitions of all such pairs u,v in Table 13.
LetQ3=∑k=1n−1/2∑l=1kCkl′rk+rn−l−3.
Now, we consider the partitions of all pairsu,v having distance 2. We have du,v=2 for all u,v∈Pi, i=1,2,…,n−1/2. We describe the partitions of all such pairs u,v in Table 14.
LetQ1′=∑k=1n−1/2ck2rk−2. If u∈Pi, v∈Pj, i,j=1,2,…,n−1/2, i≠j, then du,v=2. We describe the partitions of all such pairs u,v in Table 15.
LetQ2′=1/2∑k=1n−1/2∑l=1n−1/2cklrk+rl−2, k≠l. If u∈Pi, v∈Pj, i=1,2,…,n−3/2,j=n+1/2,n+3/2,…,n−2, i+j<n, then du,v=2. We describe the partitions of all such pairs u,v in Table 16.
LetQ3′=∑k=0n−5/2∑l=0kckl′rn−3/2−k+rn+1/2+l−3.
Using equation (10) and Theorem 6, we obtain the Schultz index of Γℤrn in Theorem 7.Table 5
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn/2+k−2,rn/2+k−2C2rn/2−k−rn/2−k−1=Fk, say, where k=0,1,2,…,n/2−1Table 6
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn/2+k−2,rn/2+l−2rn/2−k−rn/2−k−1rn/2−l−rn/2−l−1=Fkl, say, where k,l=0,1,2,…,n/2−1, k≠lTable 7
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrk−1,rn−l−2rn−k−rn−k−1rl−rl−1=Fkl′, say, where k,l satisfy the following relation: k=1,2,…,n/2−1 and for each k, l=1,2,…,kTable 8
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rk−1C2rn−k−rn−k−1=fk, say, where k=1,2,…,n/2−1Table 9
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rl−1rn−k−rn−k−1rn−l−rn−l−1=fkl, say, where k,l=1,2,…,n/2−1, k≠lTable 10
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrn/2−k−1,rn/2+l−2rn/2+k−rn/2+k−1rn/2−l−rn/2−l−1=fkl′, say, where k,l satisfy the following relation: k=1,2,…,n/2−1 and for each k, l=0,1,…,k−1Table 11
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn+1/2+k−2,rn+1/2+k−2C2rn−1/2−k−rn−3/2−k=Ck, say, where k=0,1,2,…,n−3/2Table 12
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrn+1/2+k−2,rn+1/2+l−2rn−1/2−k−rn−3/2−krn−1/2−l−rn−3/2−k=Ckl, say, where k,l=0,1,2,…,n−3/2, k≠lTable 13
Partition of pair of nodesu,v having distance 1.
du,dvCardinalityrk−1,rn−l−2rn−k−rn−k−1rl−rl−1=Ckl′, say, where k,l satisfy the following relation: k=1,2,…,n−1/2 and for each k, l=1,2,…,kTable 14
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rk−1C2rn−k−rn−k−1=ck, say, where k=1,2,…,n−1/2Table 15
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrk−1,rl−1rn−k−rn−k−1rn−l−rn−l−1=ckl, say, where k,l=1,2,…,n−1/2, k≠lTable 16
Partition of pair of nodesu,v having distance 2.
du,dvCardinalityrn−3/2−k−1,rn+1/2+l−2rn+3/2+k−rn+1/2+krn−1/2−l−rn−1/2−l−1=ckl′, say, where k,l satisfy the following relation: k=0,1,…,n−5/2 and for each k, l=0,1,…,kTheorem 7.
LetG be the zero divisor graph Γℤrn. Then, the Schultz index of G is given by(50)ScG=L1+L2+L3+2L1′+L2′+L3′,nis even,Q1+Q2+Q3+2Q1′+Q2′+Q3′,nis odd,where Li, Li′, Qi, Qi′, i=1,2,3, are described in proof of Theorem 6.Theorem 8.
LetG be the zero divisor graph Γℤrn. Then, the modified Schultz polynomial of G is given by(51)Sc∗G,x=L1+L2+L3x+L1′+L2′+L3′x2,nis even,Q1+Q2+Q3x+Q1′+Q2′+Q3′x2,nis odd,where L1=∑k=0n/2−1Fkrn/2+k−22, L2=1/2∑k=0n/2−1∑l=0n/2−1Fklrn/2+k−2rn/2+l−2, L3=∑k=1n/2−1∑l=1kFkl′rk−1rn−l−2, L1′=∑k=1n/2−1fkrk−12, L2′=1/2∑k=1n/2−1∑l=1n/2−1fklrk−1rl−1, L3′=∑k=1n/2−1∑l=0k−1fkl′rn/2−k−1rn/2+l−2, Q1=∑k=0n−3/2−1Ckrn+1/2+k−22, Q2=1/2∑k=0n−3/2∑l=0n−3/2Cklrn+1/2+k−2rn+1/2+l−2, Q3=∑k=1n−1/2∑l=1kCkl′rk−1rn−l−2, Q1′=∑k=1n−1/2ckrk−12, Q2′=1/2∑k=1n−1/2∑l=1n−1/2cklrk−1rl−1, and Q3′=∑k=0n−5/2∑l=0kckl′rn−3/2−k−1rn+1/2+l−2. Fk,Fkl,Fkl′,fk,fkl,fkl′,Ck,Ckl,Ckl′,ck,ckl, and ckl′ are illustrated in the proof of Theorem 6.Using equation (11) and Theorem 8, we obtain the modified Schultz index of Γℤrn in Theorem 9.Theorem 9.
LetG be the zero divisor graph Γℤrn. Then, the modified Schultz index of G is given by(52)Sc∗G=L1+L2+L3+2L1′+L2′+L3′,nis even,Q1+Q2+Q3+2Q1′+Q2′+Q3′,nis odd,where Li, Li′, Qi, and Qi′, i=1,2,3, are described in Theorem 8.


## 4.3. Zero Divisor GraphΓℤr×ℤs×ℤt

In this section, we obtain some distance-based, degree-distance-based, and degree-based indices of the zero divisor graphΓℤr×ℤs×ℤt, where r,s,t are primes. The structure of the zero divisor graph Γℤ5×ℤ3×ℤ2 is shown in Figure 1.Figure 1
The zero divisor graphΓℤ5×ℤ3×ℤ2.We start by computing the Hosoya polynomial in Theorem10.Theorem 10.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, its Hosoya polynomial is given by(53)HG,x=A0+A1x+A2x2+A3x3,where Aα is the number of pair of nodes having distance α.Proof.
The number of nodes and edges ofG are rs+st+rs−r−s−t and r−1s−1+r−1t−1+s−1t−1+3r−1s−1t−1. Thus, A0=rs+st+rs−r−s−t and A1=r−1s−1+r−1t−1+s−1t−1+3r−1s−1t−1. Now, we find A2 and A3. We list the information about all pairs of nodes of G having distance 2 in Table 17.
Thus, we haveA2 = 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.
Thus, we haveA3 = 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 Γℤr×ℤs×ℤt in Theorem 11.Table 17
Information related to the pair of nodes ofG having distance 2.
x,y,x,y∈VGNumber of pairsRemarks0,0,w1,0,0,w2C2t−10<w1,w2<t0,0,w1,0,vj,w2s−1t−120<w1,w2<t, 0<vj<s0,0,w1,ui,0,w2r−1t−120<w1,w2<t, 0<ui<r0,v1,0,0,v2,0C2s−10<v1,v2<s0,v1,0,ui,v2,0r−1s−120<ui<r, 0<v1,v2<su1,0,0,u2,0,0C2r−10<u1,u2<ru1,0,0,u2,vj,0r−12s−10<u1,u2<r, 0<vj<su1,0,0,u2,0,wkr−12t−10<u1,u2<r, 0<wk<tu1,v1,0,u2,v2,0C2r−1s−10<u1,u2<r, 0<v1,v2<s0,v1,w1,0,v2,w2C2s−1t−10<v1,v2<s, 0<w1,w2<tu1,0,w1,u2,0,w2C2r−1t−10<u1,u2<r, 0<w1,w2<tTable 18
Information related to the pair of nodes ofG having distance 3.
x,y,x,y∈VGNumber of pairsRemarksu1,vj,0,u2,0,wkr−12s−1t−10<u1,u2<r, 0<vj<s, 0<wk<tui,v1,0,0,v2,wkr−1s−12t−10<ui<r, 0<v1,v2<s, 0<wk<t0,vj,w1,ui,0,w2r−1s−1t−120<ui<r, 0<vj<s, 0<w1,w2<tTheorem 11.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, we have(54)WG=A1+2A2+3A3,WWG=A1+3A2+6A3,where Aα 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 Lemma1.Lemma 1.
The degrees of the vertices of the graphΓℤr×ℤs×ℤt are given by(i)
du,0,0=st−1,0<u<r(ii)
d0,v,0=rt−1,0<v<s(iii)
d0,0,w=rs−1,0<w<t(iv)
du,v,0=t−1,0<u<r,0<v<s(v)
d0,v,w=r−1,0<v<s,0<w<t(vi)
du,0,w=s−1,0<u<r,0<w<tNow, we compute theM-polynomial of the zero divisor graph Γℤr×ℤs×ℤt in the following theorem.Theorem 12.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, we have(55)MG=r−1s−1xst−1yrt−1+r−1t−1xst−1yrs−1+s−1t−1xrt−1yrs−1+r−1s−1t−1xr−1yst−1+xs−1yrt−1+xt−1yrs−1.Proof.
LetG be the total graph Γℤr×ℤs×ℤt. The edge set of G can be partitioned as follows:(56)EG=E1∪E2∪E3∪E4∪E5∪E6,where(57)E1=u,0,00,v,0∈EG:0<u<r,0<v<s,E2=u,0,00,0,w∈EG:0<u<r,0<w<t,E3=0,v,00,0,w∈EG:0<v<s,0<w<t,E4=u,0,00,v,w∈EG:0<u<r,0<v<s,0<w<t,E5=0,v,0u,0,w∈EG:0<u<r,0<v<s,0<w<t,E6=0,0,wu,v,0∈EG:0<u<r,0<v<s,0<w<t.
Also,E1=r−1s−1, E2=r−1t−1, E3=s−1t−1, and E4=E5=E6=r−1s−1t−1. From the definition, the M-polynomial of G is obtained as follows:(58)MG=∑i⩽jMi,jxiyj.
Now,(59)MG=Mst−1,rt−1xst−1yrt−1+Mst−1,rs−1xst−1yrs−1+Mrt−1,rs−1xrt−1yrs−1+Mr−1,st−1xr−1yst−1+Ms−1,rt−1xs−1yrt−1+Mt−1,rs−1xt−1yrs−1=E1xst−1yrt−1+E2xst−1yrs−1+E3xrt−1yrs−1+E4xr−1yst−1+E5xs−1yrt−1+E6xt−1yrs−1.
After putting the values ofEi, i=1,2,…,6, we obtain the required result. This completes the proof.
Now, using thisM-polynomial, we calculate some degree-based topological indices of the zero divisor graph Γℤr×ℤs×ℤt in Theorem 13 like Theorem 2.Theorem 13.
LetG be the total graph Γℤr×ℤs×ℤt. Then, we have(1)
M1G=r−1s−1rt+st−2+r−1t−1rs+st−2+s−1t−1rs+rt−2+r−1s−1t−1rs+rt+st+r+s+t−6(2)
M2G=r−1s−1rt−1st−1+r−1t−1rs−1st−1+s−1t−1rs−1rt−1+r−1s−1t−1r−1st−1+s−1rt−1+t−1rs−1(3)
FG=r−1s−1rt−12+st−12+r−1t−1rs−12+st−12+s−1t−1rs−12+rt−12+r−1s−1t−1r−12+s−12+t−12+rs−12+st−12+rt−12(4)
M2mG=r−1s−1/rt−1st−1+r−1t−1/rs−1st−1+s−1t−1/rs−1rt−1+r−1s−1t−11/r−1st−1+1/s−1rt−1+1/t−1rs−1(5)
RαG=r−1s−1rt−1st−1α+r−1t−1rs−1st−1α+s−1t−1rs−1rt−1α+r−1s−1t−1r−1st−1α+s−1rt−1α+t−1rs−1α(6)
RRαG=r−1s−1rt−1st−1−α+r−1t−1rs−1st−1−α+s−1t−1rs−1rt−1−α+r−1s−1t−1r−1st−1−α+s−1rt−1−α+t−1rs−1−α(7)
SDDG=r−1s−1rt−1/st−1+rt−1/st−1+r−1t−1rs−1/st−1+st−1/rs−1+s−1t−1rs−1/rt−1+rt−1/rs−1+r−1s−1t−1r−1/st−1+st−1/r−1+s−1/rt−1+rt−1/s−1+t−1/rs−1+rs−1/t−1(8)
HG=2r−1s−1/rt+st−2+2r−1t−1/rs+st−2+2s−1t−1/rs+rt−2+2r−1s−1t−11/st+r−2+1/rt+s−2+1/rs+t−2(9)
IG=r−1s−1rt−1st−1/rt+st−2+r−1t−1rs−1st−1/rs+st−2+s−1t−1rs−1rt−1/rs+rt−2+r−1s−1t−1r−1st−1/r+st−2+s−1rt−1/s+rt−2+t−1rs−1/t+rs−2(10)
AG=r−1s−1rt−13st−13/rt+st−43+r−1t−1rs−13st−13/rs+st−43+s−1t−1rs−13rt−13/rs+rt−43+r−1s−1t−1r−13st−13/r+st−43+s−13rt−13/s+rt−43+t−13rs−13/t+rs−43.Now, we obtain the Schultz polynomial of the zero divisor graphΓℤr×ℤs×ℤt.Theorem 14.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, the Schultz and modified Schultz polynomials of G are given by(60)ScG,x=S1x+S2x2+S3x3,Sc∗G,x=S1∗x+S2∗x2+S3∗x3,where Si and Si∗, i=1,2,3, are described in proof.Proof.
From the definition of Scultz polynomial, we have(61)ScG,x=∑u,v⊆VGdu+dvxdu,v=∑du,v=1du+dvx+∑du,v=2du+dvx2+∑du,v=3du+dvx3.
Let∑du,v=idu+dv=Si, i=1,2,3. We have from Theorem 12, S1=r-1s-1rt+st-2+s-1t-1rt+rs-2+r-1s-1rt+st-2+r-1s-1t-1rs+st+rt+r+s+t-6. Now, using Table 17 and Lemma 1, we get S2=C2t-12rs-2+s-1t-12rs+r-2+r-1t-12rs+s-2+C2s-12rt-2+r-1s-12rt+t-2+C2r-12st-2+r-12s-1st+t-2+r-12t-1st+s-2+C2r-1s-12t-2+C2s-1t-12r-2+C2r-1t-12s-2. From Table 18 and Lemma 1, we get S3=r−12s−1t−1s+t−2+r−1s−12t−1r+t−2+r−1s−1t−12r+s−2. Putting the values of Si, i=1,2,3 in equation (61), we obtain the required result. From the definition of Scultz polynomial, we have(62)Sc∗G,x=∑u,v⊆VGdudvxdu,v=∑du,v=1dudvx+∑du,v=2dudvx2+∑du,v=3dudvx3.
Let∑du,v=idudv=Si∗, i=1,2,3. We have from Theorem 12, S1*=r-1s-1rt-1st-1+s-1t-1rs-1st-1+r-1s-1rt-1rs-1+r-1s-1t-1st-1r-1+rt-1s-1+rs-1t-1. Now using Table 17 and Lemma 1, we get S2*=C2t-1rs-12+s-1t-12rs-1r-1+r-1t-12rs-1s-1+C2s-1rt-12+r-1s-12rt-1t-1+C2r-1st-12+r-12s-1st-1t-1+r-12t-1st-1s-1+C2r-1s-1t-12+C2s-1t-1r-12+C2r-1t-1s-12. From Table 18 and Lemma 1, we get S3∗=3r−12s−12t−12. Putting the values of Si, i=1,2,3 in equation (62), we obtain the required result.
Now, using equations (10) and (11), we obtain the Schultz and modified Schultz indices in Theorem 15.Theorem 15.
LetG be the zero divisor graph Γℤr×ℤs×ℤt. Then, the Schultz and modified Schultz indices of G are given by(63)ScG=S1+2S2+3S3,Sc∗G=S1∗+2S2∗+3S3∗,where Si and Si∗, i=1,2,3, are described in proof of Theorem 14.


## 5. Conclusion

We obtained some degree-based, degree-distance-based, and distance-based topological indices of the total graphTℤnn∈ℤ+, the zero divisor graph Γℤrn (r is prime, n∈ℤ+), and the zero divisor graph Γℤr×ℤs×ℤt (r,s,t are primes) through algebraic polynomials. Firstly, we computed the explicit expressions of algebraic polynomials for the aforesaid structures. Then, topological indices are recovered from the polynomials. The results can play an important role in visualizing the topology of the graphs under consideration.



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