Abstract
The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo psqt for all distinct primes p, q and . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph ΓE(R) are derived for .
Introduction
The study of graphs defined on algebraic structures has been an active topic of research in the last few decades. The main question in the area is to recognize finite rings through the properties of various graphs defined on it. The notion of the zero-divisor graph of a commutative ring was introduced by I. Beck in (Beck, 1988), where he considered the set of zero divisors including zero and introduced the concepts such as diameter, grith and clique number of a zero divisor graph. Then later on in (Anderson and Livingston, 1999), Anderson and Livingston changed the vertex set of the zero-divisor graph, they considered only the vertices of the non-zero zero-divisors. For more details, one may see the survey (Singh and Bhat, 2020) and the references therein for the vast literature on the study of zero-divisor graphs.
The Wiener index is one of the important graph indices, and has a variety of applications in pharmaceutical science and in the structure of nanotubes. For results and applications of Wiener index, see (Devillers and Balaban, 1999; Dobrynin et al., 2001; Dehmer and Emmert-Streib, 2014; Dobrynin and Iranmanesh, 2020). There are some works of the Wiener index were done for the ring of integers modulo n. Let us review some of the work done on the topological indices of the zero-divisor graphs. Let p, q be distinct prime numbers. Ahmadi et al. (Ahmadi and Nezhad, 2011) in 2011 has provided an algorithm to determining the Wiener index of for n = p2, pq. In 2018, Mohammad et al. (Mohammad and Authman, 2018) has extended the result by determining the Wiener index of a zero-divisor graph of for n = pm and pmq, where and m ≥ 2 using the Hosoya polynomial. Pirzada et al. (Pirzada et al., 2020) in 2020 determined the Wiener index of a zero-divisor graph and a compressed zero-divisor of for . In (Asir and Rabikka, 2021), recently a constructed method to calculate the Wiener index of zero-divisor graph of for any positive integer n is determined. The authors of (Asir and Rabikka, 2021) calculated the complete formula through restrict n as product of distinct primes and the remaining cases. In 2022, Selvakumar et al. (Selvakumar et al., 2022) visualized the zero-divisor graph Γ(R) as a generalized composition of suitable choices of graphs and derived a formula for the Wiener index of the graph .
In this paper, we are interested in the parameter Wiener index of graphs for the rings of integers modulo psqt. Although the formulas in the general case for the rings of have been obtained in literatures (Asir and Rabikka, 2021) and (Selvakumar et al., 2022), compared with their results, our formula is more direct and convenient for calculation the Wiener index . We also get the formula for compressed zero-divisor graph.
Preliminaries
Throughout this paper we assume that R denotes a commutative ring with identity, Z(R) be its set of zero-divisors, the (nonempty) set of nonzero zero-divisors and unit elements denoted by Z(R)* and U(R). We use to note the ring of integers.
Definition 1LetGbe a graph and letuandvbe two vertices ofG. The distance betweenuandv, denoted bydG(u, v), is defined to be the length of the shortest path betweenuandv. The Wiener index of the graphG, denoted byW(G), is defined to be the sum of all distanced between any two vertices ofG.Let dG(v) denote the sum of distances of the vertex v from all the vertices of G, then the Wiener index can be redefined asLet R be an arbitrary finite commutative ring with unity. We define an equivalence relation ∼ on Z(R)* as follows. For x, y ∈ Z(R)*, define x ∼ y if and only if ann(x) = ann(y) where ann(x) = {r ∈ R|rx = 0}. We call these classes the equiv-annihilator classes of the zero-divisor graph Γ(R).We write d (x, y) to denote the distance between x and y in Z(R)*, and write x ∼ y to denote x and y are adjacent, otherwise x ≁ y. Let U, V be subsets of the vertex of Γ(R), the U ↔ V shall denote that each vertex of U is adjacent to every vertex of V, and U ↮ V denotes that no vertex of U is adjacent to every vertex of V.The so-called compressed zero-divisor graph of a ring was first defined by the Spiroff et al. in (Spiroff and Wickham, 2011).
Definition 2For a commutative ringRwith 1 ≠ 0, a compressed zero-divisor graph of a ringRis the undirected graph ΓE(R) with vertex setZ(RE) − [0] = RE − {[0], [1]} defined byRE = {[x]|x ∈ R}, where [x] = {y ∈ R|ann(x) = ann(y)} and two distinct vertices [x] and [y] are adjacent if and only if [x][y] = [0] = [xy], that is, if and only ifxy = 0.In what follows, we use the graph-theoretic notions from (Douglas, 2001).
Main results
In this section, we first give a structure of using the method of equivalence classification.
Let p, q be distinct prime numbers and , the vertex set of be divided into disjoint subsets V00, …, Vij, …, Vst, where
We noted that Vst =∅ and . For the convenience of presentation, we always assumes that V00 and Vst are empty sets in the following, unless otherwise specified. Therefore
Example 1Consider the ring. The vertex set ofisIt is not difficult to see that Vij be the equiv-annihilator classes of , where 0 ≤ i ≤ s and 0 ≤ j ≤ t. If i < s and j < t, for any x, y ∈ Vij. Let z ∈ ann(x), then z = k′ps−iqt−j. So yz = (kpiqj) (k′ps−iqt−j) = kk′psqt, that is, z ∈ ann(y). If i < s and j = t, for any x, y ∈ Vij. Let z ∈ ann(x), then z = k′ps−i. So yz = (kpiqt) (k′ps−i) = kk′psqt, that is, z ∈ ann(y). If i = s and j < t, for any x, y ∈ Vij. Let z ∈ ann(x), then z = k′qt−j. So yz = (kpsqj) (k′qt−j) = kk′psqt, that is, z ∈ ann(y). Thus ann(x) = ann(y) for any x, y ∈ Vij.Next, we prove some elementary properties of the vertex subsets Vij.
Lemma 1
,
q, letn=
psqtfor someandVijbe the equiv-annihilator classes ofwhere0 ≤
i≤
sand0 ≤
j≤
t. Then(1)
(2) Vij ↔ Vi′j′if and only ifi + i′ ≥ sandj + j′ ≥ t.
Proof(1) we consider the following cases.Case 1:i ≠ s and j ≠ t.Let Sij be the set of all the elements that can be divisible by piqj in . By the inclusion-exclusion principle,Note that |Sij| = |{kpiqj|0 ≤ k < ps−iqt−j}| = ps−iqt−j. SinceAndThenCase 2:i = s.SinceThenCase 3:j = t.Since
Then
(2) Let x = kijpiqj ∈ Vij, y = ki′j′pi′qj′ ∈ Vi′j′. If i + i′ ≥ s and j + j′ ≥ t, then
Proposition 1For distinct prime numbersp, q, letfor some. Thend(x, y) = 1, 2 or 3.
Proof. LetV01, V10, ⋯ , Vs,t−1, Vs−1,t be the equiv-annihilator classes of , where Vij defined by (1). For and , where 0 ≤ i1, i2 ≤ s and 0 ≤ j1, j2 ≤ t.If i1 + i2 ≥ s and j1 + j2 ≥ s, then x ∼ y and d (x, y) = 1 by lemma 1. So we only need to consider the cases of i1 + i2 < s or j1 + j2 < s in the following, that is, x ≁ y. Without loss of generality,we may assume that i1 + i2 < s. Consider the following cases.Case 1: 0 < i1, i2 < s.Let i = s − min{i1, i2}, j = t. We have i1 + i ≥ s and j1 + j ≥ t, also i + i2 ≥ s and j + j2 ≥ t. Then . Hence, d (x, y) = 2.Case 2:i1 = 0 and i2 = 0.Let i = s, j = t − min{j1, j2}. We have i1 + i ≥ s and j1 + j ≥ t, also i + i2 ≥ s and j + j2 ≥ t. Then . Hence, d (x, y) = 2.Case 3:i1 = 0 and i2 ≠ 0. Consider the following subcases.Subcase3.1: If j2 = 0. Let i3 = s, i4 = s − i2, j3 = t − j1, and j4 = t. We haveAndThus .SinceThen and . Therefore, d (x, y) = 3.Subcase3.2: If j2 ≠ 0. Let i = s and j = t − min{j1, j2}. We haveAndThus . Therefore, d (x, y) = 2.Case 4:i1 ≠ 0 and i2 = 0. A similar argument as in Case 3 shows that d (x, y) = 2 or 3.We have already shown that in any case, d (x, y) = 1, 2 or 3. Now, we can calculate the Wiener index of .
For distinct prime numbersp, q, and some. The Wiener indexwhere
Proof. Letn = psqt, we have V01, V10, …, Vs−1,t, Vs,t−1 is the partition of ,where Vij defined by (1). For any two different elements x, y in Vij. By the proof of Proposition 1, there are the following cases.Case 1: or .In this case, we have d (x, y) = 2. ThenCase 2: and .In this case, d (x, y) = 1. ThenLet x and y be the elements in the two different equiv-annihilator classes, Vij and Vi′j′, respectively. Consider the following cases.Case 3:i + i′ ≥ s and j + j′ ≥ t.By Lemma 1, d (x, y) = 1. ThenCase 4: 0 < i + i′ < s or 0 < j + j′ < t.Subcase 4.1:i = 0 and j′ = 0.In this case, we have d (x, y) = 3. HenceSubcase 4.2:i′ = 0 and j = 0.In this case, d (x, y) = 3. HenceSubcase 4.3: If i, j′ are not both equal to 0, and i′, j are not both equal to 0.In this case, d (x, y) = 2. HenceIn conclusion, the Weiner index isTherefore the result holds, by Lemma 1. The following Table gives the exact value of for n = 2s3t, where 1 ≤ s ≤ 3 and 1 ≤ t ≤ 3.The compressed zero-divisor graph of can be obtained by treating the set Vij, 0 ≤ i ≤ s, 0 ≤ j ≤ t, as a single vertex. To illustrate, let’s give an example in the following.
Example 2Consider the ring, the vertex set ofis divided into 10 setsV01,V02,V03,V10,V11,V12,V13,V20,V21,V22. Then the associated compressed zero-divisor graphis shown inFigure 1.Before proving the next result we need the following lemma.
FIGURE 1
the compressed zero-divisor graph .
Lemma 2
,
q, letn=
psqtfor someandbe the compressed zero-divisor graph of. Then(1) V(G) = {Vij|0 ≤ i ≤ s, 0 ≤ j ≤ t} .
(2)
Proof(1) Note thatwhere be the units set of .Let , such that ann(x) = ann(y). Assume that (i, j) ≠ (i′, j′). Without loss of generality, we may let i < i′. There are the following cases.Case1:i < i′ < s.Since z = ups−i′qt ∈ ann(y). But xz = u1ups−i′+iqt+j is not divisible by n, a contradiction. therefore, (i, j) = (i′, j′) and [x] = [y] = Vij.Case 2:i < s < i′.Since z = ups−i−1qt ∈ ann(y). But xz = u1ups−1qt+j is not divisible by n, a contradiction. therefore, (i, j) = (i′, j′) and [x] = [y] = Vij.Case 3:s < i < i′.In this case, we have j < t and j′ < t. If j ≠ j′, then z = uqmin{t−j,t−j′} ∈ ann(x) or z = uqmin{t−j,t−j′} ∈ ann(y) but not both. A contradiction. therefore, j = j′ and [x] = [y] = Vsj.
Then the result is holds.
(2) Let denote the sum of distances of the vertex Vij from the vertices of G with a distance of k, where k = 1, 2 or 3 by Proposition 1. Then
Remark 1From the above lemma, it can be easily seen that the cardinalities of the vertex set ofG, that is, |V(G)| = (s + 1) (t + 1) − 2. Soas shown in Example 1.The following theorem gives the Wiener index of .
For distinct prime numbersp, q, and some. The Wiener index of the compressed zero-divisor graphis
Proof. Let n = psqt, and . we have V01, V10, …, Vs−1,t, Vs,t−1 are all the vertices of G by Lemma 2, where Vij defined by (1). Then
Example 3Consider the ring. The Wiener index of the compressed zero-divisor graphisBy Theorem 2.
Conclusion
In this paper, we have described the structure of the graph for all distinct primes p, q and by partition of the vertex set. Consider the partition of the vertex set into the subsets V01, V10, …, Vij, ⋯ , Vs−1,t, Vs,t−1 as seen (1). Then Vij ↔ Vi′j′ if and only if i + i′ ≥ s and j + j′ ≥ t. Based on this structure, we proved that the distance of two vertices of are contained in the set {1, 2, 3}, and derived an explicit formula for Wiener index of the graph in Theorem 1 using the basic counting principles.
In addition, we run the formula obtained through MATLAB software and get the data in Table 1. Then, we studied the structure of the compressed zero-factor graph of by treating the set Vij as a single vertex of the compressed zero-divisor graph . We showed that the degree of vertex Vij generally includes six cases, with the number of the vertices of the graph be (s + 1) (t + 1) − 2. Finally we derive the corresponding formula for Wiener index in Theorem 2. Of course, we can also implement it in software if needed.
TABLE 1
| 2 × 3 | 22 × 3 | 23 × 3 | 2 × 32 | 22 × 32 | 23 × 32 | 2 × 33 | 22 × 33 | 23 × 33 | |
|---|---|---|---|---|---|---|---|---|---|
| 4 | 38 | 210 | 109 | 504 | 2294 | 1267 | 5152 | 22136 |
The Wiener index of for n = 2s3t.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
RL conceived of the presented idea, and WY developed the theory and performed the computations, verified the analytical methods. RL investigated and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.
Funding
The authors are very grateful to the referee for careful reading of the manuscript and helpful suggestions. This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), Hechi University Research Fund for Advanced Talents (2019GCC005) and Hechi University Research Fund (2018XJQN007).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Summary
Keywords
wiener index, zero-divisor graphs, compressed zero-divisor graph, residue class rings, equivalence classification
Citation
Wei Y and Luo R (2022) The wiener index of the zero-divisor graph for a new class of residue class rings. Front. Chem. 10:985001. doi: 10.3389/fchem.2022.985001
Received
03 July 2022
Accepted
04 August 2022
Published
13 September 2022
Volume
10 - 2022
Edited by
Xiyun Zhang, Jinan University, China
Reviewed by
Hongjie Bi, Okinawa Institute of Science and Technology Graduate University, Japan
Wei Wang, Chongqing Medical University, China
Updates
Copyright
© 2022 Wei and Luo.
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*Correspondence: Ricai Luo, hcxylor@126.com
This article was submitted to Theoretical and Computational Chemistry, a section of the journal Frontiers in Chemistry
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.