# Computing Wiener and Hyper-Wiener Indices of Zero-Divisor Graph ofℤℊ3×ℤI1I2

Authors: Yonghong Liu; Abdul Rauf; AdnanAslam; Saira Ishaq; Abudulai Issa
Journal: Journal of Function Spaces
 (2022)

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## Abstract

LetS=ℤℊ3×ℤI1I2 be a commutative ring where ℊ,I1andI2 are positive prime integers with I1≠I2. The zero-divisor graph assigned to S is an undirected graph, denoted as YS with vertex set V(Y(S)) consisting of all Zero-divisor of the ring S and for any c, d ∈V(Y (S)), cd∈EYS if and only if cd =0. A topological index/descriptor is described as a topological-invariant quantity that transforms a molecular graph into a mathematical real number. In this paper, we have computed distance-based polynomials of YR i-e Hosoya polynomial, Harary polynomial, and the topological indices related to these polynomials namely Wiener index, and Hyper-Wiener index.


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## Body

## 1. Introduction

The Zero-divisor graph is the undirected graph on the set of the Zero-divisors of a commutative ring. In a Zero-divisor graph, the set of zero-divisors are considered as vertices and pairs of elements whose product is zero as its edges. The Zero-divisor graph is very helpful to study the algebraic properties of rings using graph-theoretical tools.In 1988 Beck was the first who gives the idea of interlinking two main mathematics topics Algebra, and Graph theory [1]. First, he presented the concept of a Zero-divisor graph of Commutative Ring R, in which all the elements of ring R were considered as the vertices of a Zero-divisor graph, and those two distinct vertices c and d are connected if and only if cd =0. Beck’s main objective in his work was to show the coloring of the Commutative ring. Naseer and Anderson extended the work by Beck’s in [2]. In [3], Anderson-Livingston worked on the Zero-divisor graphs in which only non-zero Zero-divisors are considered as the vertices. Anderson-Livingston discussed the relations between ring theoretic properties of R and graph-theoretic properties of the Y(R). Furthermore, this study presents some important results of zero-divisor graphs. Later Anderson, Frazier, Lauve, Levy, Livingston, and Shapiro [4, 5] worked on translating the algebraic properties of rings into graphical language.Redmond developed the idea of the Zero-divisor graphs associated to non-commutative rings [6]. He defined a Zero-divisor graph associated with a non-commutative ring in many more ways, including both directed and undirected graphs. Redmond [7] carried on this work by extending the concept of Zero-divisor graph of a Commutative ring to an ideal-based Zero-divisor graph of a Commutative ring by replacing elements whose product is zero with elements whose product lies in some ideal I of ring R. Since then, many researchers have worked on it and defined graphs such as unit graphs, the equivalence class of Zero-divisor graphs, total graphs, ideal-based Zero-divisor graphs, the Jacobson graphs, and so on (see, [7–11]).Let G be graph with vertex set V(G) and edge set E(G). The degree of a vertex v is the number of edges attached to v. For any two vertices u and v, the distance between them is denoted by d(u,v) and is defined as the length of the shortest path joining them. For instance, d (u, v) = 0 If and only if u = v and d (u, v) = ∞ if there does not exist any possible shortest path. For more details on the basic definition related to graph theory, the readers can see the book by [12].Let Z(S) be the set of Zero-divisors of the Commutative ring ‘S’. The Zero-divisor graph of S is denoted byY(S) = (V(Y (S)), E(Y (S))), which is an undirected graph with V(Y (S)) = Z(S) \ {0} as vertex set, and for different c, d ∈ V(Y(S)), the vertices c and d have an edge cd∈EYS between them if and only if cd = 0 [13]. It is very interesting to translate the algebraic properties of algebraic graphs into numerical molecular descriptors. A topological index/descriptor is described as a topological-invariant quantity that transforms a molecular graph into a mathematical real number. QSPR/QSAR studies are majorly concerned with the application of these molecular descriptors or Topological indices. Many molecular descriptors have been introduced in the last decade, demonstrating their importance. A molecular structure is denoted as a graph with atoms as vertices and bonds as edges. Then, using various forms of topological indices, one can study both the algebraic and chemical aspects of the compounds. There are two main types of topological indices, the first based on degree [14–19] and the second based on distance. Randic Connectivity index, Zagreb Indices, the Harmonic index, Atom bond Connectivity, and Geometric Arithmetic index are degree-based topological indices. Some well-known distance-based topological indices are the Wiener index, Hosoya index, and Estrada index [20, 21]. For details related to the computation of Hosoya and Harary polynomial of zero divisor graphs associated to some commutative rings, the readers can see [22–24].Eccentricity-based indices have been effectively used to construct a variety of mathematical models for the prediction of biological activities of various types. Several authors have examined the uses and mathematical properties of these indices. Further, the readers can see [25–28] for more details about eccentricity based-indices. In 1988 Hosoya polynomial was introduced by Haruo Hosoya [29]. With its vast application in graph theory and chemistry, it is proved to be an effective distance-based topological index [30]. The Hosoya polynomial has many chemical applications [31]. Almost all distance-based topological indices can be computed from this polynomial [32–34]. The Hosoya polynomial is related to a variety of topological indices, the most well-known of which is the Wiener, Hyper Wiener, and Harary polynomial. For further study see [21, 35–37]. Hosoya polynomial of a graph H is defined as:
(1)HH,x=∑vϵVH∑uϵVHxdu,v.The Harary polynomial was introduced in 1985 and is denoted byhH,x and is defined as:
(2)hH,x=∑vϵVH∑uϵVHxdu,vdu,v.The generalized Harary index of graph H is denoted byhtH and defined as:
(3)htH=∑vϵVH∑uϵVH1du,v+t.Where t= 1,2, 3, .... The Wiener index is the oldest topological index introduced by Harold Wiener in 1947 for the study of boiling points of paraffin [38]. It plays a very important role in inverse structure-property relationship problems [39]. For applications and mathematical properties of Weiner index see [40–46]. The Wiener index is defined as:
(4)WH=12∑vϵVH∑uϵVHdu,vIn 1993 another distance-based topological index was introduced by Randic known as the Hyper-Wiener index [35]. This index is used for predicting physicochemical properties of organic compounds [47], and is defined as:
(5)WWH=12∑vϵVH∑uϵVHdu,v2+du,vIt is easy to observe that there is an effective relation between Hosoya Polynomial, Wiener Index, and Hyper-Wiener Index.(6)WH=H′H:1,WWH=H′H;1+12H′′H;1


## 2. Result and Discussion

Letℤℊ3×ℤI1I2 be a commutative ring where ℊ,I1andI2 are positive prime integers withI1≠I2. The total number of elements in this ring isℊ3I1I2. We are only concerned with the set of zero-divisors of ℤℊ3×ℤI1I2 denoted by VYℤℊ3×ℤI1I2 having cardinalityℊ3−ℊ2I1+I2−1+ℊ2I1I2−1. We made the degree-based disjoint partition for vertices of the zero-divisor graph Yℤℊ3×ℤI1I2 of the commutative ring, which are given as:
(7)W1=0,v:v∈ZI1I2withv≠0,k1I1,k2I2,1≤k1≤I2−1,1≤k2≤I1−1,(8)W2=0,v:v=k1I1,1≤k1≤I2−1,(9)W3=0,v:v=k2I2,1≤k2≤I1−1,(10)W4=u,0:uϵℤℊ3,withu≠0,t1ℊ,1≤t1≤ℊ2−1,(11)W5=u,v:u≠0,t1ℊ,1≤t1≤ℊ2−1,v=k1I1,1≤k1≤I2−1,(12)W6=u,v:u≠0,t1ℊ,1≤t1≤ℊ2−1,v=k2I2,1≤k2≤I1−1,(13)W7=u,0:u=t1ℊ,1≤t1≤ℊ2−1witht1≠lℊ,1≤t1≤ℊ−1,(14)W8=u,v:u=t1ℊ,1≤t1≤ℊ2−1witht1≠lℊ,1≤t1≤ℊ−1,v≠0,k1I1,k2I2,1≤k1≤I2−1,1≤k2≤I1−1,(15)W9=u,v:u=t1ℊ,1≤t1≤ℊ2−1witht1≠lℊ,1≤t1≤ℊ−1,v=k1I1,1≤k1≤I2−1,(16)W10=u,v:u=t1ℊ,1≤t1≤ℊ2−1witht1≠lℊ,1≤t1≤ℊ−1,v=k1I2,1≤k2≤I1−1,(17)W11=u,0:u=t2ℊ2,1≤t2≤ℊ−1,(18)W12=u,v:u=t2ℊ2,1≤t2≤ℊ−1,v≠0,k1I1,k2I2,1≤k1≤I2−1,1≤k2≤I1−1,(19)W13=u,v:u=t2ℊ2,1≤t2≤ℊ−1,v=k1I1,1≤k1≤I2−1,,(20)W14=u,v:u=t2ℊ2,1≤t2≤ℊ−1,vk2I2,1≤k2≤I1−1.From the above, it is clear thatVYℤℊ3×ℤI1I2=⋃i=114Wi. Observe that W1=I1−1I2−1,W2=I2−1,W3=I1−1,W4=ℊ3−ℊ2,W5=ℊ3−ℊ2I2−1,W6=ℊ3−ℊ2I1−1,W7=ℊ2−ℊ, W8=ℊ2−ℊI1I2−I1−I2+1, W9=ℊ2−ℊI2−1, W10=ℊ2−ℊI1−1, W11=ℊ−1, W12=ℊ−1I1I2−I1−I2+1, W13=ℊI2−ℊ−I2+1, W14=ℊI1−ℊ−I1+1. Furthermore dv=ℊ3−1 for vϵW1, dv=I1ℊ3−1 for vϵW2, dv=I1ℊ3−1 for vϵW3, dv=I1I2−1 for vϵW4, dv=I1−1 for vϵW5, dv=I2−1 for vϵW6, dv=ℊI1I2−1 for vϵW7,dv=ℊ−1 for vϵW8,dv=I1ℊ−1 for vϵW9,dv=I1ℊ−1 for vϵW10,dv=ℊ2I1I2−1 for vϵW11,dv=ℊ2−1 for vϵW12,dv=I1ℊ2−1 for vϵW13,dv=I2ℊ2−1 for vϵW14.Let1≤i≤14be a fixed integer andu∈Wi. From the above partition one can observe that the distance du,v is same for any v∈Wj, where 1≤j≤14. This observation is depicted in Figure 1. If there is an edge between any two partition sets Wi and Wj, then it means that du,v=1 for any u∈Wiand v∈Wj. It can be seen that the maximum distance between any two vertices of Zero-divisor graph Yℤℊ3×ℤI1I2 is atmost three. This important fact is stated in Lemma 1.Lemma 1.
The maximum diameter of the Zero-divisor graphYS of a commutative ring is 3 [3].Figure 1
Zero-divisor graph ofℤℊ3×ℤI1I2.We have summarized the eccentricity and degree of each vertex of Zero-divisor graphYℤℊ3×ℤI1I2 in Table 1:Table 1
Eccentricity and degree of each vertex of Zero-divisor graphYℤℊ3×ℤI1I2.
VerticesEccentricityDegreeFrequencyW13ℊ3−1I1I2−I1−I2+1W22I1ℊ3−1I2−1W32I2ℊ3−1I1−1W43I1I2−1ℊ3−ℊ2W53I1−1ℊ3−ℊ2I2−1W63I2−1ℊ3−ℊ2I1−1W72ℊI1I2−1ℊ2−ℊW83ℊ−1ℊ2−ℊI1−1I2−1W93I1ℊ−1ℊ2−ℊI2−1W103I2ℊ−1ℊ2−ℊI1−1W112ℊ2I1I2−1ℊ−1W123ℊ2−1ℊ−1I1I2−I1−I2+1W133I1ℊ2−1ℊI2−ℊ−I2+1W143I2ℊ2−1ℊI1−ℊ−I1+1


## 3. Hosoya and Harary Polynomial of Zero Divisor GraphYℤℊ3×ℤI1I2

Now we compute the Harary and Hosoya polynomial of zero divisor graphYℤℊ3×ℤI1I2. First, we find the distance du,v for each pair of vertices u,v of Yℤℊ3×ℤI1I2. For this, we consider the partitionsWi, 1≤i≤14 as vertices. Now we compute the distance du,v between any two vertices such that u∈Wiandv∈Wj. Let x∈ℤ,x>0 and DPxWi,Wj=u,v∈Wi,Wjdu,v=x. If i=j, we simply use the notation DPxWi. The cardinality of the set of ordered pair of vertices which are adjacent to each other is denoted by DP1Yℤℊ3×ℤI1I2.Lemma 2.
The cardinality of the set of ordered pair of vertices inYℤℊ3×ℤI1I2 which are adjacent to each other is:
(21)DP1Yℤℊ3×ℤI1I2=ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−142Proof.
By Handshaking lemma and using table1 we concluded
(22)DP1Yℤℊ3×ℤI1I2=12ℊ3−1+I1ℊ3−1+I2ℊ3−1+I1I2−1+I1−1+I2−1+ℊI1I1−1+ℊ−1+I1ℊ−1+I2ℊ−1+ℊ2I1I2−1+ℊ2−1+I1ℊ2−1+I2ℊ2−1=ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−142Lemma 3.
The cardinality of the set of ordered pair of vertices inYℤℊ3×ℤI1I2 which are distance 2 to each other is
(23)DP2Yℤℊ3×ℤI1I2=Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7.
Where(24)Ω1=I1I2−I1−I2!+I2−2!+I1−2!+ℊ3−ℊ2−1!+ℊ3−ℊ2I2−1−1!+ℊ3−ℊ2I1−1−1!+ℊ2−ℊ−1!+ℊ2−ℊI1I2−I1−I2+1−1!+ℊ12I2−ℊ12−ℊI2+ℊ−1!+ℊ2−ℊI1−1−1!+ℊ−2!+ℊ−1I1I2−I1−I2+1−1!+ℊI1−ℊ−I1!+ℊI2−ℊ−I2!,(25)Ω2=I1I2−I1−I2+1I1I2−1ℊ2−1+I1+I2,(26)Ω3=I2−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1+I1−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1,(27)Ω4=ℊ3−ℊ2I1+I2−2ℊ3−ℊ2+I1I2ℊ2−ℊ+ℊ−1+I1+I2+1−I1−I2,(28)Ω5=ℊ3−ℊ2I2−1ℊ2−ℊ+ℊ2−ℊ1+I1I2−I1−I2+ℊ2−ℊI2−1+ℊ−1+ℊ−1I2−1+ℊ3−ℊ2I1−1ℊ2−ℊ+ℊ2−ℊI1−1+ℊ2−ℊI2−1+ℊ−1+ℊ−1I1−1,(29)Ω6=ℊ2−ℊ2I1I2−1+ℊ2−ℊI1I2−I1−I2+1I1+I2ℊ2−ℊ+ℊ−11+I1I2−2ℊ2+ℊ2I2−ℊ2−ℊI2+ℊℊ2I1−ℊ2−2ℊI1+ℊI1I2+ℊ−I1I2+I1,(30)Ω7=ℊ2I1−ℊ2−ℊI1+ℊℊI1I2−I1I2−ℊ+1+ℊI1−ℊ−I1+1I2−1I1+I2−2.Proof.
From Figure1. We conclude that W1 is at distance 2 from W1,W2,W3,W8,W9,W10,W12,W13andW14. Hence by using the values from table 1, we have:
(31)DP2W1=I1I2−I1−I2!(32)DP2W1,W2=I1−1I2−12(33)DP2W1,W3=I1−12I2−1(34)DP2W1,W8=ℊ2−ℊI1−12I2−12(35)DP2W1,W9=ℊ2−ℊI1−1I2−12(36)DP2W1,W10=ℊ2−ℊI1−12I1−1(37)DP2W1,W12=ℊ−1I1−12I2−12(38)DP2W1,W13=ℊ−1I1−1I2−12(39)DP2W1,W14=ℊ−1I1−12I2−1
From Figure1, we conclude that W2 is at distance 2 from W2,W5,W8,W9,W12andW13. Hence by using the values from table 1, we have:
(40)DP2W2=I2−2!(41)DP2W2,W5=ℊ3−ℊ2I2−12(42)DP2W2,W8=ℊ2−ℊI1−1I2−12(43)DP2W2,W9=ℊ2−ℊI2−12(44)DP2W2,W12=ℊ−1I1−1I2−12(45)DP2W2,W13=ℊ−1I2−12
From Figure1, we conclude that W3 is at distance 2 from W3,W6,W8,W10,W12andW14. Hence by using the values from table 1, we have:
(46)DP2W3=I1−2!(47)DP2W3,W6=ℊ3−ℊ2I1−12(48)DP2W3,W8=ℊ2−ℊI1−12I2−1(49)DP2W3,W10=ℊ2−ℊI1−12(50)DP2W3,W12=ℊ−1I1−12I2−1(51)DP2W3,W14=ℊ−1I1−12
From Figure1. We conclude that W4 is at distance 2 from W4,W5,W6,W7,W8,W9,W10,W11,W13andW14. Hence by using the values from table 1, we have:
(52)DP2W4=ℊ3−ℊ2−1!(53)DP2W4,W5=ℊ3−ℊ22I2−1(54)DP2W4,W6=ℊ3−ℊ22I1−1(55)DP2W4,W7=ℊ2−ℊℊ3−ℊ2(56)DP2W4,W8=ℊ3−ℊ2ℊ2−ℊI1−1I2−1(57)DP2W4,W9=ℊ3−ℊ2ℊ2−ℊI2−1(58)DP2W4,W10=ℊ3−ℊ2ℊ2−ℊI1−1(59)DP2W4,W11=ℊ3−ℊ2ℊ−1(60)DP2W4,W13=ℊ3−ℊ2ℊ−1I2−1(61)DP2W4,W14=ℊ3−ℊ2ℊ−1I1−1
From Figure1, we conclude that W5is at distance 2 from W5,W7,W8,W9,W11andW13. Hence by using the values from table 1, we have:
(62)DP2W5=ℊ3−ℊ2I2−1−1!(63)DP2W5,W7=ℊ3−ℊ2ℊ2−ℊI2−1(64)DP2W5,W8=ℊ3−ℊ2ℊ2−ℊI1−1I2−12(65)DP2W5,W9=ℊ3−ℊ2ℊ2−ℊI2−12(66)DP2W5,W11=ℊ3−ℊ2ℊ−1I2−1(67)DP2W5,W13=ℊ3−ℊ2ℊ−1I2−12
From Figure1, we conclude that W6 is at distance 2 from W6,W7,W10,W11andW14. Hence by using the values from table 1, we have:
(68)DP2W5=ℊ3−ℊ2I1−1−1!(69)DP2W5,W7=ℊ3−ℊ2ℊ2−ℊI1−1(70)DP2W6,W10=ℊ3−ℊ2ℊ2−ℊI1−12(71)DP2W6,W11=ℊ3−ℊ2ℊ−1I1−1(72)DP2W6,W14=ℊ3−ℊ2ℊ−1I1−12
From Figure1, we conclude that W7 is at distance 2 from W7,W8,W9andW10. Hence by using the values from table 1, we have:
(73)DP2W7=ℊ2−ℊ−1!(74)DP2W7,W8=ℊ2−ℊ2I1−1I2−1(75)DP2W7,W9=ℊ2−ℊ2I2−1(76)DP2W7,W10=ℊ2−ℊ2I1−1
From Figure1, we conclude that W8 is at distance 2 from W8,W9,W10,W12,W13andW14. Hence by using the values from table 1, we have:
(77)DP2W8=ℊ2−ℊI1−1I2−1−1!(78)DP2W8,W9=ℊ2−ℊ2I1−1I2−12(79)DP2W8,W10=ℊ2−ℊ2I1−12I2−1(80)DP2W8,W12=ℊ2−ℊ2ℊ−1I1−12I2−12(81)DP2W8,W13=ℊ2−ℊ2ℊ−1I1−1I2−12(82)DP2W8,W14=ℊ2−ℊ2ℊ−1I1−12I2−1
From Figure1, we conclude that W9 is at distance 2 from W9,W10,W12,andW13. Hence by using the values from table 1, we have:
(83)DP2W9=ℊ2−ℊI2−1−1!(84)DP2W9,W10=ℊ2−ℊ2I1−1I2−1(85)DP2W9,W12=ℊ2−ℊI2−12ℊ−1I1−1(86)DP2W9,W13=ℊ2−ℊI2−12ℊ−1
From Figure1, we conclude that W10 is at distance 2 from W10,W12,W13,andW14. Hence by using the values from table 1, we have:
(87)DP2W10=ℊ2−ℊI1−1−1!(88)DP2W10,W12=ℊ2−ℊI1−12I2−1ℊ−1(89)DP2W10,W13=ℊ2−ℊI1−1I2−1ℊ−1(90)DP2W10,W14=ℊ2−ℊI1−12ℊ−1
From Figure1, we conclude that W11 is at distance 2 from W11. Hence by using the values from table 1, we have:
(91)DP2W11=ℊ−2!
From Figure1, we conclude that W12 is at distance 2 from W12,W13,andW14. Hence by using the values from table 1, we have:
(92)DP2W12=ℊ−1I1−1I2−1−1!(93)DP2W12,W13=ℊ−12I1−1I2−12(94)DP2W12,W14=ℊ−1I1−12I2−1
From Figure1, we conclude that W13 is at distance 2 from W13. Hence by using the values from table 1, we have:
(95)DP2W13=ℊ−1I2−1−1!
From Figure1, we conclude that W14 is at distance 2 from W14. Hence by using the values from table 1, we have:
(96)DP2W14=ℊ−1I2−1−1!
Finally, we add all the values computed in equations (31)–(96), and we get the required formula for DP2Yℤℊ3×ℤI1I2:
(97)DP2Yℤℊ3×ℤI1I2=I1I2+I1+I2!+I2−2!+I1−2!+ℊ3−ℊ2−1!+ℊ3−ℊ2I2−1−1!+ℊ3−ℊ2I1−1−1!+ℊ2−ℊ−1!+ℊ2−ℊI1I2−I1−I2+1−1!+ℊ12I2−ℊ12−ℊI2+ℊ−1!+ℊ2−ℊI1−1−1!+ℊ−2!+ℊ−1I1I2−I1−I2+1−1!+ℊI1−ℊ−I1!+ℊI2−ℊ−I2!+I1I2−I1−I2+1I1I2−1ℊ2−1+I1+I2+I2−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1+I1−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊ−1I1−1+ℊ−1+ℊ3−ℊ2I1+I2−2ℊ3−ℊ2+I1I2ℊ2−ℊ+ℊI1+I2−1−I1−I2+1+ℊ3−ℊ2I2−1ℊ2−ℊ+ℊ2−ℊI1−1I2−1+ℊ12I2−ℊ12−ℊI2+ℊ+ℊ−1+ℊ−1I2−1+ℊ3−ℊ2I1−1ℊ2−ℊ+ℊ12I1−ℊ12−ℊI1+ℊ+ℊ12I2−ℊ12−ℊI2+ℊ+ℊ−1+ℊ−1I1−1+ℊ2−ℊ2I1I2−1+ℊ2−ℊ1+I1I2−I1−I2I1+I2ℊ2−ℊ+ℊ−11+I1I2−2ℊ2+ℊ2I2−ℊ2−ℊI2+ℊℊ2I1−ℊ2−2ℊI1+ℊI1I2+ℊ−I1I2+I1+ℊ2I1−ℊ2−ℊI1+ℊ1−ℊ+ℊI1I2−I1I2+ℊ−1I1−1I2−1I1+I2−2=Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7Lemma 4.
The cardinality of the set of ordered pair of vertices inYℤℊ3×ℤI1I2 which are distance 3 to each other is
(98)DP3Yℤℊ3×ℤI1I2=ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3Proof.
By using the figure, we conclude that:(99)DP3Yℤℊ3×ℤI1I2=W1W5+W6+W4W12+W5W6+W10+W12+W14+W6W8+W9+W12+W13
Now by putting values from table1, we obtain:
(100)DP3Yℤℊ3×ℤI1I2=I1I2−I1−I2+1ℊ3−ℊ2I2−1+ℊ3−ℊ2I2−1+ℊ3−ℊ2ℊI1−ℊ−I1+1I2−1+ℊ3−ℊ2I1I2−I1−I2+1ℊ3−ℊ2+ℊ2−ℊI1−1+ℊI1−ℊ−I1+1I2−1+ℊI1−ℊ−I1+1+ℊ3−ℊ2I1−1ℊ2−ℊI1I2−I1−I2+1+ℊ2−ℊI2−1+ℊI1−ℊ−I1+1I2−1+ℊI2−ℊ−I2+1=ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3Theorem 5.
The Hosoya polynomialHYℤℊ3×ℤI1I2 of the graph Yℤℊ3×ℤI1I2 is:
(101)HYℤℊ3×ℤI1I2=ℊ3−ℊ2I1+I2−1+ℊ2I1I2−1+ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−142x+Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7x2+ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3x3
Where(102)Ω1=I1I2+I1+I2!+I2−2!+I1−2!+ℊ3−ℊ2−1!+ℊ3−ℊ2I2−1−1!+ℊ3−ℊ2I1−1−1!+ℊ2−ℊ−1!+ℊ2−ℊI1I2−I1−I2+1−1!+ℊ12I2−ℊ12−ℊI2+ℊ−1!+ℊ2−ℊI1−1−1!+ℊ−2!+ℊ−1I1I2−I1−I2+1−1!+ℊI1−ℊ−I1!+ℊI2−ℊ−I2!,(103)Ω2=I1I2−I1−I2+1I1I2−1ℊ2−1+I1+I2,(104)Ω3=I2−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1+I1−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1,(105)Ω4=ℊ3−ℊ2I1+I2−2ℊ3−ℊ2+I1I2ℊ2−ℊ+ℊ−1+I1+I2+1−I1−I2,(106)Ω5=ℊ3−ℊ2I2−1ℊ2−ℊ+ℊ2−ℊ1+I1I2−I1−I2+ℊ2−ℊI2−1+ℊ−1+ℊ−1I2−1+ℊ3−ℊ2I1−1ℊ2−ℊ+ℊ2−ℊI1−1+ℊ2−ℊI2−1+ℊ−1+ℊ−1I1−1,(107)Ω6=ℊ2−ℊ2I1I2−1+ℊ2−ℊI1I2−I1−I2+1I1+I2ℊ2−ℊ+ℊ−11+I1I2−2ℊ2+ℊ2I2−ℊ2−ℊI2+ℊℊ2I1−ℊ2−2ℊI1+ℊI1I2+ℊ−I1I2+I1,(108)Ω7=ℊ2I1−ℊ2−ℊI1+ℊℊI1I2−I1I2−ℊ+1+ℊI1−ℊ−I1+1I2−1I1+I2−2.Proof.
The Hosoya polynomial of zero divisor graphYℤℊ3×ℤI1I2 can be calculated by using lemma 2, lemma 3 and lemma 4 as follows:
(109)HYℤℊ3×ℤI1I2,x=DP0Yℤℊ3×ℤI1I2+DP1Yℤℊ3×ℤI1I2x+DP2Yℤℊ3×ℤI1I2x2+DP3Yℤℊ3×ℤI1I2x3=ℊ3−ℊ2I1+I2−1+ℊ2I1I2−1+ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−142x+Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7x2+ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3x3Theorem 6.
The Harary polynomialhYℤℊ3×ℤI1I2,x of the graph Yℤℊ3×ℤI1I2 is
(110)hYℤℊ3×ℤI1I2,x=3ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−14x+3Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7x2+2ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3x3
Where(111)Ω1=I1I2+I1+I2!+I2−2!+I1−2!+ℊ3−ℊ2−1!+ℊ3−ℊ2I2−1−1!+ℊ3−ℊ2I1−1−1!+ℊ2−ℊ−1!+ℊ2−ℊI1I2−I1−I2+1−1!+ℊ12I2−ℊ12−ℊI2+ℊ−1!+ℊ2−ℊI1−1−1!+ℊ−2!+ℊ−1I1I2−I1−I2+1−1!+ℊI1−ℊ−I1!+ℊI2−ℊ−I2!,(112)Ω2=I1I2−I1−I2+1I1I2−1ℊ2−1+I1+I2,(113)Ω3=I2−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1+I1−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1,(114)Ω4=ℊ3−ℊ2I1+I2−2ℊ3−ℊ2+I1I2ℊ2−ℊ+ℊ−1+I1+I2+1−I1−I2,(115)Ω5=ℊ3−ℊ2I2−1ℊ2−ℊ+ℊ2−ℊ1+I1I2−I1−I2+ℊ2−ℊI2−1+ℊ−1+ℊ−1I2−1+ℊ3−ℊ2I1−1ℊ2−ℊ+ℊ2−ℊI1−1+ℊ2−ℊI2−1+ℊ−1+ℊ−1I1−1,(116)Ω6=ℊ2−ℊ2I1I2−1+ℊ2−ℊI1I2−I1−I2+1I1+I2ℊ2−ℊ+ℊ−11+I1I2−2ℊ2+ℊ2I2−ℊ2−ℊI2+ℊℊ2I1−ℊ2−2ℊI1+ℊI1I2+ℊ−I1I2+I1,(117)Ω7=ℊ2I1−ℊ2−ℊI1+ℊℊI1I2−I1I2−ℊ+1+ℊI1−ℊ−I1+1I2−1I1+I2−2.Proof.
The Harary polynomial ofYℤℊ3×ℤI1I2 can be calculated by using lemma 2, lemma 3 and lemma 4 as follows:
(118)hYℤℊ3×ℤI1I2,x=6DP1Yℤℊ3×ℤI1I2x+3DP2Yℤℊ3×ℤI1I2x2+2DP3Yℤℊ3×ℤI1I2x3=3ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−14x+3Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7x2+2ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3x3Corollary 7.
The Wiener index and Hyper-Wiener index for zero divisor graphYℤℊ3×ℤI1I2 can be expressed as:
(119)WYℤℊ3×ℤI1I2=ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−142+Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7+3ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3(120)WWYℤℊ3×ℤI1I2=ℊ3+ℊ2+ℊ+ℊ3I1+I2+I1I2+I1+I2ℊ2+ℊ+1−14+3Ω1+Ω2+Ω3+Ω4+Ω5+Ω6+Ω7+6ℊ3−ℊ2I1−1I2−1ℊ3+I1ℊ+I2ℊ2−3
Where(121)Ω1=I1I2+I1+I2!+I2−2!+I1−2!+ℊ3−ℊ2−1!+ℊ3−ℊ2I2−1−1!+ℊ3−ℊ2I1−1−1!+ℊ2−ℊ−1!+ℊ2−ℊI1I2−I1−I2+1−1!+ℊ12I2−ℊ12−ℊI2+ℊ−1!+ℊ2−ℊI1−1−1!+ℊ−2!+ℊ−1I1I2−I1−I2+1−1!+ℊI1−ℊ−I1!+ℊI2−ℊ−I2!,(122)Ω2=I1I2−I1−I2+1I1I2−1ℊ2−1+I1+I2,(123)Ω3=I2−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1+I1−12ℊ3−ℊ2+ℊ2−ℊI1−1+ℊ2−ℊ+ℊI1−ℊ−I1+1+ℊ−1,(124)Ω4=ℊ3−ℊ2I1+I2−2ℊ3−ℊ2+I1I2ℊ2−ℊ+ℊ−1+I1+I2+1−I1−I2,(125)Ω5=ℊ3−ℊ2I2−1ℊ2−ℊ+ℊ2−ℊ1+I1I2−I1−I2+ℊ2−ℊI2−1+ℊ−1+ℊ−1I2−1+ℊ3−ℊ2I1−1ℊ2−ℊ+ℊ2−ℊI1−1+ℊ2−ℊI2−1+ℊ−1+ℊ−1I1−1,(126)Ω6=ℊ2−ℊ2I1I2−1+ℊ2−ℊI1I2−I1−I2+1I1+I2ℊ2−ℊ+ℊ−11+I1I2−2ℊ2+ℊ2I2−ℊ2−ℊI2+ℊℊ2I1−ℊ2−2ℊI1+ℊI1I2+ℊ−I1I2+I1,(127)Ω7=ℊ2I1−ℊ2−ℊI1+ℊℊI1I2−I1I2−ℊ+1+ℊI1−ℊ−I1+1I2−1I1+I2−2.


## 4. Conclusion

In this paper, we computed some distance-based polynomial of zero divisor graphYℤℊ12ℊ2×ℤI1I2 namely Hosoya polynomial, Harary polynomial, Wiener index, and hyper Wiener index. We have seen that the eccentricity of any vertex of a zero divisor graph is always 2 or 3, which helps for the computation of the above polynomial. These computations can be helpful in computation of eccentricity based topological indices of the considered zero divisor graph.



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*Source: 2046173-2022-09-19.xml*