# Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

Authors: Ali Ahmad; S. C. López
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

LetR be a commutative ring with nonzero identity and let ZR be its set of zero divisors. The zero-divisor graph of R is the graph ΓR with vertex set VΓR=ZR∗, where ZR∗=ZR\0, and edge set EΓR=x,y:x⋅y=0. One of the basic results for these graphs is that ΓR is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤp2q2, namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.


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## Body

## 1. Introduction

Algebraic structures have been investigated significantly for their nearby connection with representation theory and number theory; likewise, they have been widely concentrated in combinatorics [1, 2]. Despite the expansive theoretical research in these areas, restricted rings and fields got consideration for their applications to cryptography and coding theory.In mathematical chemistry, a graphical structure of a chemical compound is a representation of the structural formula. In a chemical graph, vertices and edges represent the atoms and their chemical bonds of the compound, respectively. Molecular descriptors for a particular chemical compound are calculated on basis of the corresponding molecular graph. A topological index is a graph invariant that is obtained from it. In [3], the first topological index, namely, the Wiener index, was introduced. Nowadays, it is widely used in QSAR (“Quantitative Structure Activity Relationship”), whose properties are surveyed in [4, 5].Topological indices are classified as degree based [6–9] and distance based of graphs. Some well-known topological indices based on the degrees of a graph are the Randić connectivity index, Zagreb indices, Harmonic index, atom bond connectivity, and geometric arithmetic index. The Wiener index, Hosaya index, and Estrada index are distance-based topological indices [10, 11]. Topological indices formulate the criteria for the development of compound structures, and numerical activities on these structures extend multidisciplinary research. In what follows, we cite some of them.A relationship among the stability of linear alkanes and the branched alkanes is examined using theABC index, which helped in computation of strain energy for cycle alkanes [12, 13]. The GA index is more appropriate and efficient to correlate certain physico-chemical characteristics for predictive power than the Randić connectivity index [14, 15]. The Zagreb indices are powerful tools for the calculation of total p-electron energy of the molecules with precise approximation [16]. The degree-based topological indices are more useful to examine the chemical characteristics of distinct molecular structures. Eccentricity-based topological indices are useful as a key for the judgement of toxicological, physico-chemical, and pharmacological properties of a compound through the structure of its molecules. The study of the QSAR is known for this sort of analysis [17]. By exploring [18, 19], further applications of topological indices can be obtained.


### 1.1. Distance-Based Topological Indices and Polynomials

In this section, we introduce the topological indices and polynomials that will be obtained for the graphs studied in this paper. We recall some concepts from graph theory. LetG be a (undirected) graph. If there is a path between any two distinct vertices of G, then G is a connected graph. For two distinct vertices x,y∈VG, we denote dx,y the length of a shortest path connecting x and y (dx,x=0 and, dx,y=∞ if no such a path exists). The diameter of the graph G is the maximum length of the shortest path connecting two distinct vertices of G, that is, diamG=maxdx,y:x≠y∈VG. The number of edges incidence a vertex x of simple graph G is called the degree of the vertex x, denoted as dx.The Wiener index [11] was introduced by Wiener in 1947 to illustrate the connection between physico-chemical properties of organic compounds and the index of their molecular graphs:(1)WG=12∑u∈VG∑v∈VGdu,v.Randić [20] and Randić et al. [21] introduced a modified version of the Wiener index that is used for predicting physico-chemical properties of organic components. The new index was called the hyper-Wiener index and it is defined as follows:(2)WWG=12∑u∈VG∑v∈VGdu,v+du,v2.The Hosoya polynomial was introduced in 1989 [22]. The definition is as follows:(3)HG,x=12∑u∈VG∑v∈VGxdu,v.Dobrynin and Kochetova [23], and independently, Gutman [24] introduced a degree distance index, which is known as the Schultz index. Let G be a connected graph and du be the degree of u∈VG. Then, the Schultz index or the degree distance of G is defined as follows:(4)ScG=12∑u∈VG∑v∈VGdu+dvdu,v.Klavžar and Gutman defined, in [25], the modified Schulz index of a graph as follows:(5)Sc∗G=12∑u∈VG∑v∈VGdu⋅dvdu,v.Finally, Gutman, in [24], introduced two topological polynomials, namely, the Schulz polynomial ScG,x and the modified Schulz polynomial Sc∗G,x as follows:(6)ScG,x=12∑u∈VG∑v∈VGdu+dvxdu,v,Sc∗G,x=12∑u∈VG∑v∈VGdu⋅dvxdu,v.The connection between the above polynomials and the previous two indices is stated below:(7)ScG=∂ScG,x∂xx=1,Sc∗G=∂Sc∗G,x∂xx=1.


### 1.2. Zero-Divisor Graphs

LetR be a commutative ring with nonzero identity and let ZR be its set of zero divisors. The zero-divisor graph of R is the graph ΓR with vertex set VΓR=ZR∗, where ZR∗=ZR∖0, and edge set EΓR=x,y:x⋅y=0. As usual, an edge x,y is simply denoted as xy. Zero-divisor graphs were introduced by Beck [2] in 1988 and then studied by Anderson and Naseer in [26]. These authors were interested in colorings and the original definition included all elements in R, even the zero. Later on, Anderson and Livingston [27] made emphasis on the relationship between ring-theoretical properties and graph-theoretical properties and reformulated the definition as it appears in the lines above. One of the basic results in this relationship is the following one.Theorem 1.
(see [27]). Let R be a commutative ring. Then, ΓR is connected with diameter less or equal to 3.The study conducted in [28, 29] may serve as a survey that is very interesting to find the relation between ring-theoretic properties and graph-theoretic properties of ΓG. Some applications and relation between algebraic theory and chemical graph theory can be seen in [1, 18, 30]. In this paper, we presented some results that interplay in the relation between a zero-divisor graph and chemical graph theory. The structure of the paper is as follows. In Section 2, we describe the family of zero-divisor graphs of the form Γℤp2q2, where p and q are different primes, and we also count pairs of vertices that are exactly at distance i, for i=1,2,3. In Section 3, we obtain the Wiener index and the Hosoya, the Shultz, and the modified Shultz polynomials of Γℤp2q2. We also obtain the Shultz and the modified Shultz indices of Γℤp2q2.


## 1.1. Distance-Based Topological Indices and Polynomials

In this section, we introduce the topological indices and polynomials that will be obtained for the graphs studied in this paper. We recall some concepts from graph theory. LetG be a (undirected) graph. If there is a path between any two distinct vertices of G, then G is a connected graph. For two distinct vertices x,y∈VG, we denote dx,y the length of a shortest path connecting x and y (dx,x=0 and, dx,y=∞ if no such a path exists). The diameter of the graph G is the maximum length of the shortest path connecting two distinct vertices of G, that is, diamG=maxdx,y:x≠y∈VG. The number of edges incidence a vertex x of simple graph G is called the degree of the vertex x, denoted as dx.The Wiener index [11] was introduced by Wiener in 1947 to illustrate the connection between physico-chemical properties of organic compounds and the index of their molecular graphs:(1)WG=12∑u∈VG∑v∈VGdu,v.Randić [20] and Randić et al. [21] introduced a modified version of the Wiener index that is used for predicting physico-chemical properties of organic components. The new index was called the hyper-Wiener index and it is defined as follows:(2)WWG=12∑u∈VG∑v∈VGdu,v+du,v2.The Hosoya polynomial was introduced in 1989 [22]. The definition is as follows:(3)HG,x=12∑u∈VG∑v∈VGxdu,v.Dobrynin and Kochetova [23], and independently, Gutman [24] introduced a degree distance index, which is known as the Schultz index. Let G be a connected graph and du be the degree of u∈VG. Then, the Schultz index or the degree distance of G is defined as follows:(4)ScG=12∑u∈VG∑v∈VGdu+dvdu,v.Klavžar and Gutman defined, in [25], the modified Schulz index of a graph as follows:(5)Sc∗G=12∑u∈VG∑v∈VGdu⋅dvdu,v.Finally, Gutman, in [24], introduced two topological polynomials, namely, the Schulz polynomial ScG,x and the modified Schulz polynomial Sc∗G,x as follows:(6)ScG,x=12∑u∈VG∑v∈VGdu+dvxdu,v,Sc∗G,x=12∑u∈VG∑v∈VGdu⋅dvxdu,v.The connection between the above polynomials and the previous two indices is stated below:(7)ScG=∂ScG,x∂xx=1,Sc∗G=∂Sc∗G,x∂xx=1.


## 1.2. Zero-Divisor Graphs

LetR be a commutative ring with nonzero identity and let ZR be its set of zero divisors. The zero-divisor graph of R is the graph ΓR with vertex set VΓR=ZR∗, where ZR∗=ZR∖0, and edge set EΓR=x,y:x⋅y=0. As usual, an edge x,y is simply denoted as xy. Zero-divisor graphs were introduced by Beck [2] in 1988 and then studied by Anderson and Naseer in [26]. These authors were interested in colorings and the original definition included all elements in R, even the zero. Later on, Anderson and Livingston [27] made emphasis on the relationship between ring-theoretical properties and graph-theoretical properties and reformulated the definition as it appears in the lines above. One of the basic results in this relationship is the following one.Theorem 1.
(see [27]). Let R be a commutative ring. Then, ΓR is connected with diameter less or equal to 3.The study conducted in [28, 29] may serve as a survey that is very interesting to find the relation between ring-theoretic properties and graph-theoretic properties of ΓG. Some applications and relation between algebraic theory and chemical graph theory can be seen in [1, 18, 30]. In this paper, we presented some results that interplay in the relation between a zero-divisor graph and chemical graph theory. The structure of the paper is as follows. In Section 2, we describe the family of zero-divisor graphs of the form Γℤp2q2, where p and q are different primes, and we also count pairs of vertices that are exactly at distance i, for i=1,2,3. In Section 3, we obtain the Wiener index and the Hosoya, the Shultz, and the modified Shultz polynomials of Γℤp2q2. We also obtain the Shultz and the modified Shultz indices of Γℤp2q2.


## 2. The Zero-Divisor Graph onΓℤp2q2

Let us start by introducing some notation that will be used along the paper. We assume thatp and q are different positive primes.Lemma 1.
Let0≤i,j≤2. Let Ai,j=piqj⊂ℤp2q2, for 0≤i+j≤4 and Ai,j=∅, otherwise. Then,(i)
Ai,j=p2−iq2−j(ii)
Ai+1,j∪Ai,j+1⊂Ai,j(iii)
Ai+1,j∩Ai,j+1=Ai+1,j+1, whenever i+j+2≤4
The vertices ofΓℤp2q2 can be split into blocks such that all vertices in the same block have the same behavior. From this partition, we can easily describe the structure of Γℤp2q2, that is, the content of the following lemma.Lemma 2.
For0≤i,j≤2 and 0<i+j<4, let Bi,j=Ai,j∖Ai+1,j∪Ai,j+1. Then,(8)Bi,j=p2−iq2−j−p1−iq2−j−p2−iq1−j+p1−iq1−j,ifmaxi,j=1,q2−j−q1−j,ifi=2,p2−i−p1−i,ifj=2.Moreover,(i)
Ifi,j≥1, then ΓBi,j is a clique. Otherwise, ΓBi,j is a set of independent vertices.(ii)
LetBi,j and Bi′,j′ such that i+i′=j+j′=2. Then, the edges uv, u∈Bi,j and v∈Bi′,j′, define a complete bipartite graph.According to the definition, all vertices in the same block of the zero-divisor graph onℤp2q2 have the same degree. For more details on this graph, see [30]. Let dij be the degree of any vertex in Bi,j. Following the notation of the previous lemma, we also conclude the following information.Lemma 3.
For0≤i,j≤2 and 0<i+j<4, then(9)dij=piqj−i−j,ifmaxi,j=1,p2qj−j−1,ifi=2,piq2−i−1,ifj=2.Figure1 shows the structure of the zero-divisor graph ℤp2q2. White vertices represent blocks of independent vertices in ℤp2q2, whereas black vertices represent cliques.The structure shown in Lemmas2 and 3 can be completed by showing the distance between pairs of vertices, which only depends on the block they belong to. This information appears in Table 1.LetTPiG, i∈ℤ and i>0, the number of pairs of vertices at distance i in a graph G.Figure 1
The structure ofℤp2q2.Table 1
Distance between pair of vertices, according to the block they belong to 1.
B1,0B0,1B2,0B1,1B0,2B1,2B2,1B1,02322312B0,13232221B2,02322112B1,12221211B0,23212221B1,21211211B2,12121111Lemma 4.
LetΓℤp2q2 be a zero-divisor graph; then,(10)TP1Γℤp2q2=3pqp−1q−1+12pq−1pq−2.Proof.
The size ofΓℤp2q2 is given by(11)B1,2B2,0+B1,0+B2,1B0,2+B0,1+B2,0B0,2+EKB1,2+B1,1+B2,1.
That is, by introducing the values described in Lemma2, the result follows.Lemma 5.
LetΓℤp2q2 be a zero-divisor graph; then,(12)TP2Γℤp2q2=12pqpqp2+q2−6+p+q+2.Proof.
The number of pairs of vertices at distance 2 is given by the formula that follows:(13)TP2Γℤp2q2=B2,0+B1,02+B0,2+B0,12+B1,1B2,0+B1,0+B0,2+B0,1+B2,1B2,0+B1,0+B1,2B0,2+B0,1,that is,(14)TP2Γℤp2q2=B2,0+B1,0B2,0+B1,0−12+B1,1+B2,1+B0,2+B0,1B0,2+B0,1−12+B1,1+B1,2.
Thus, by Lemma2, we obtain the following expression:(15)TP2Γℤp2q2=pqpq+2q−1−12q−1+qp+2p−1−12p−1.
Hence, after simplification, the result follows.Lemma 6.
LetΓℤp2q2 be a zero-divisor graph; then,(16)TP3Γℤp2q2=pqp−1q−1pq−1.Proof.
The number of pairs at distance exactly 3,TP3, is given by the following expression:(17)TP3Γℤp2q2=B0,1B2,0+B0,1B1,0+B0,2B1,0.
Thus, by Lemma2, we get the result.


## 3. Distance-Based Topological Indices and Polynomials ofΓℤp2q2

Now, we are ready to state and prove the following theorems.Theorem 2.
The Winner index ofΓℤp2q2 is(18)WΓℤp2q2=3p2q2p−1q−1+pqpqp2+q2−6+p+q+2+12pq−1pq−2.Proof.
The diameter ofΓℤp2q2 is 3. Thus, there are pairs of vertices at distance 1,2, and 3, and the Winner index can be obtained as follows:(19)WΓℤp2q2=TP1+2TP2+3TP3.
By Lemmas4–6, we get TP1, TP2, and TP3, respectively. Thus, by introducing these values in (19), we get, after simplification, the required result.Lemma 7.
The Hosoya polynomial ofΓℤp2q2 is p2q2−1/2+TP1x+TP2x2+TP3x3.Theorem 3.
The Hosoya polynomial ofΓℤp2q2 is(20)HΓℤp2q2,x=p2q2−12+3pqp−1q−1+12pq−1pq−2x+12pqpqp2+q2−6+p+q+2x2+pqp−1q−1pq−1x3.Proof.
The result follows by Lemma7 and by Lemmas 4–6.Lemma 8.
The Hyper–Wiener index ofΓℤp2q2 is 2TP1+6TP2+12TP3.Theorem 4.
The Hyper–Wiener index ofΓℤp2q2 is(21)WWΓℤp2q2=12p2q2p−1q−1+pqpq3p2+3q2−23+9p+9q−3+2.Proof.
The result follows by Lemma8 and by Lemmas 4–6, after some simplifications.Lemma 9.
Letαij=Bi,jdij, where dij is the degree of any vertex in Bi,j, 0≤i,j≤2 and 0<i+j<4. Then, the Schultz polynomial of Γ=Γℤp2q2=V,E is equal to(22)ScΓ,x=α10B0,1+B0,2+α01B1,0+B2,0+α20B0,1+α02B1,0x3+B0,1+B0,2α12+α11+B1,0+B2,0α21+α11+d12−B1,2α20+α10+d21−B2,1α02+α01x2+∑u∈VΓdu2x+2EΓ.Proof.
Since the diameter ofΓ is 3, 3 is by definition the degree of ScΓ,x. The 3rd coefficient of the polynomial is given by(23)B0,1B1,0d10+d01+B2,0B0,1d20+d01+B0,2B1,0d02+d10.
That is,α10B0,1+α01B1,0+α20B0,1+α01B2,0+α02B1,0+α10B0,2. From this expression, we clearly obtain the 3rd coefficient that appears in the statement.
The coefficients ofx2 and x (see Table 1) are given, respectively, by(24)s2=2B1,02d10+2B2,02d20+2B0,12d01+2B0,22d02+B2,0B1,0d20+d10+B1,1B1,0d11+d10+B1,1B0,1d11+d01+B1,1B2,0d11+d20+B0,2B0,1d02+d01+B0,2B1,1d02+d11+B1,2B0,1d12+d01+B1,2B0,2d12+d02+B2,1B1,0d21+d10+B2,1B2,0d21+d20,s1=2B1,12d11+2B1,22d12+2B2,12d21+B1,2B2,0d12+d20+B2,1B0,2d21+d02+B1,2B1,0d12+d10+B2,1B0,1d21+d01+B0,2B2,0d02+d20+B1,1B1,2d11+d12+B1,1B2,1d11+d21+B2,1B1,2d21+d12.
By doing similar transformations as above, we obtain the2n and the 1st coefficient, respectively, that appears in the statement. Finally, the independent term appears when we apply the formula to each vertex.Theorem 5.
Letp and q be different primes. Then, the Schultz polynomial of Γ=Γℤp2q2 is equal to(25)ScΓ,x=pp−1qq−12p2q+2pq2−4pq−p2−q2+2x3+pq4p3q2+4p2q3−3p3q−12p2q2−3pq3+2pq2+2p2q+6pq+2p2+2q2−4x2+p−1q−1p3q+p2q2+pq3+2p2q+2pq2−10pq+4+p−1pq2−22+q−1pq2−22x+p−1q−15pq−2+p−1pq2−2+q−1p2q−2.Proof.
Consider the formula obtained in Lemma9. Note that(26)∑u∈VΓdu2=α20d20+α02d02+α10d10+α01d01+α11d11+α12d12+α21d21and 2EΓ=α20+α02+α10+α01+α11+α12+α21. By introducing the values of Bi,j, dij (collected in Lemmas 2 and 3), and αij in terms of p and q, the result follows.Corollary 1.
The Schultz index or the degree distance ofΓ=Γℤp2q2 is equal to(27)ScΓ,x=3pp−1qq−12p2q+2pq2−4pq−p2−q2+2+2pq4p3q2+4p2q3−3p3q−12p2q2−3pq3+2pq2+2p2q+6pq+2p2+2q2−4+p−1q−1p3q+p2q2+pq3+2p2q+2pq2−10pq+4+p−1pq2−22+q−1pq2−22.Lemma 10.
Letαij=Bi,jdij, where dij is the degree of any vertex in Bi,j, 0≤i,j≤2 and 0<i+j<4. Then, the modified Schultz polynomial of Γ=Γℤp2q2=V,E is equal to(28)Sc∗Γ,x=α10α01+α20α01+α02α10x3+α10α10−d102+α20α20−d202+α01α01−d012+α02α02−d022+α20α10+α11α10+α11α01+α11α20++α02α01+α02α11+α12α01+α12α02+α21α10+α21α20x2+α11α11−d112+α12α12−d122+α21α21−d212+α12α20+α21α02+α12α10+α21α01+α02α20+α11α12+α11α21+α21α12x+∑u∈Vdu2.Proof.
Since the diameter ofΓ is 3, 3 is by definition the degree of Sc∗Γ,x. The 3rd coefficient of the polynomial is given by(29)B0,1B1,0d10d01+B2,0B0,1d20d01+B0,2B1,0d02d10,that is, α10α01+α20α01+α02α10.
The coefficients ofx2 and x (see Table 1) are given, respectively, by(30)s2∗=B1,02d102+B2,02d202+B0,12d012+B0,22d022+B2,0B1,0d20d10+B1,1B1,0d11d10+B1,1B0,1d11d01+B1,1B2,0d11d20+B0,2B0,1d02d01+B0,2B1,1d02d11+B1,2B0,1d12d01+B1,2B0,2d12d02+B2,1B1,0d21d10+B2,1B2,0d21d20,s1∗=B1,12d112+B1,22d122+B2,12d212+B1,2B2,0d12d20+B2,1B0,2d21d02+B1,2B1,0d12d10+B2,1B01d21d01+B0,2B2,0d02d20+B1,1B1,2d11d12+B1,1B2,1d11d21+B2,1B1,2d21d12.
By doing similar transformations as above, we obtain the2n and the 1st coefficient, respectively, that appears in the statement. Finally, the independent term appears when we apply the formula to each vertex.Theorem 6.
Letp and q be different primes. Then, the modified Schultz polynomial of Γ=Γℤp2q2 is equal to(31)Sc∗Γ,x=pp−12qq−123pq−p−q−1x3+p−12q−128p2q2+p2q+pq22−p2−q2−11pq−p−q+12p−1q−16p3q3−2p3q−2pq3−6p2q2−4p2q−4pq2−p3−q3−3p2−3q2+10pq+5p+5qx2+p−1q−113p3q3−5p3q2−2p3q−p2q3−8p2q2−8p2q−6pq3−4pq2+14pq+4p+4q+12p−1p−2pq2−22+12q−1q−2p2q−22x+p−1pq2−22+q−1pq2−22+p−1q−1p3q+p2q2+pq3+p2q+pq2−8pq+4.Proof.
Consider the formula obtained in Lemma10. Recall that ∑u∈VΓdu2=α20d20+α02d02+α10d10+α01d01+α11d11+α12d12+α21d21. By introducing the values of Bi,j, dij (collected in Lemmas 2 and 3), and αij in terms of p and q, the result follows.Corollary 2.
The modified Schultz index ofΓ=Γℤp2q2 is equal to(32)ScΓ,x=p−12q−1225p2q2−2p2q−2pq2−p2−q2−14pq−p−q+p−1q−119p3q3−5p3q2−4p3q−p3−14p2q2−12p2q−3p2−8pq3−8pq2+24pq+9p−q3−3q2+9q+12p−1p−2pq2−22+12q−1q−2p2q−22.


## 4. Conclusion

The structure of zero-divisor graphs of the form ofΓℤp2q2 is particularly interesting for studying distance-based topological indices. First, because its diameter is exactly 3, but also because there are defined blocks, with a complete bipartite connection between them, of either independent vertices or complete graphs (cliques). In this paper, we have focused on the Wiener index and the Hosoya, the Shultz, and the modified Shultz polynomials of Γℤp2q2 and, finally, on the Shultz and the modified Shultz indices of Γℤp2q2. For that reason, we have introduced some notation that could be useful not only for studying other distance-base topological indices of Γℤp2q2 but also for other graphs of the form Γℤpmqn, for m,n positive integers. A key point in this notation is the study of pairs that are exactly a distance one (the size of the graph), two, or three. We think that a possible line of future research should include the study of paths connecting pairs of vertices a different distances and the extension to other indices, as for instance, the Estrada index.



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