Wiener index of graphs over rings: a survey

Abstract

This article presents a survey of results consisting of the Wiener index of graphs associated with commutative rings. In particular, we focus on zero-divisor graphs, unit graphs, total graphs and prime graphs. Further, we have posed some open problems corresponding to the graphs considered.

Keywords:

• Wiener index
• unitary Cayley graph
• unit graph
• total graph

2000 Mathematics Subject Classification:

• Primary: 05C25
• Secondary: 15A18
• 92E10
• 13M05

1. Introduction

Topological indices have received special attention in mathematical chemistry. These are invariants which can be calculated from the underlying molecular graphs and exhibit good correlations with physical and chemical properties of the corresponding molecules. Molecules and molecular compounds are often modeled by the molecular graph. A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Topological indices are used for correlation analysis in theoretical chemisty, pharmacology, toxicology, and environmental chemistry. There are two major classes of topological indices namely distance-based topological indices and degree-based topological indices of graphs. Among these, the classes of distance-based topological indices play a vital role in chemical graph theory. One of the well studied distance-based topological index is called Wiener index of graph.

1.1. Wiener index of graphs

The Wiener index was introduced by Harold Wiener [38] in 1947. Wiener index has remarkable variety of chemical applications. Wiener himself used it to predict boiling point of parafin. He named this index as path number. Later on, the path number was renamed as the Wiener index. The Wiener index is considered as one of the most used topological indices which highly correlates many physical and chemical properties of molecular compounds. In particular, the Wiener index has a variety of applications in pharmaceutical science and in the structure of nanotubes. For results and applications of Wiener index, see [14,17]. Based on this interest on Wiener index some other topological indices have also been discovered. The mathematical representation of Wiener index was given by Hosoya in [18].

The distance between vertices u, v of a graph G, denoted by $d_G(u,v),$ is the length of the shortest path in G beginning at u and ending at v. Then the Wiener index of a connected graph G is defined by, $$W(G)=\frac{1}{2}\sum _{u\in V(G)}d(u|G)$$ where $d(u|G)$ is the distance of a vertex u of a graph, $$d(u|G)=\sum _{v\in V(G)}d(u,v).$$

The hyper-Wiener index of acyclic graphs was introduced by Milan Randic in 1993 [31], as a generalization of the Wiener index. Then, in 1995, Klein et al. [20] generalized the hyper-Wiener index concept to all the connected graphs. The hyper-Wiener index of a graph G is defined as $$HW(G)=\frac{1}{2}\sum _{u,v\in V(G)}{d}_G(u,v)+\frac{1}{2}\sum _{u,v\in V(G)}{d}_G^2(u,v)$$

1.2. Graphs associated to rings

In the past three decades, graphs constructed from algebraic structures have been studied extensively by many authors and have become a major field of research. The idea of constructing a graph from an algebraic structure was introduced by Arthur Cayley in 1878. He constructed a graph from groups. Another important graph construction is the construction of graphs from rings. The study of graphs from rings contributes to the interplay between the ring structure and the derived graph structure. One can sometimes translate algebraic properties of commutative rings to graph-theoretic language, and then the geometric properties of the graphs can help explore some interesting results related to commutative rings. The study of graphs from rings starts from the most well-studied zero-divisor graphs from commutative rings. The other well-studied graphs on the subject concentrated in this article are total graphs, unit graphs and prime graphs. For more details on graphs from rings, one may refer [4].

1.3. Notation

Throughout the article, R will be a commutative ring with identity $1\ne 0.$ We denote $Z(R), U(R)$ and Nil(R) to indicate the set of zero-divisors of R, the set of units of R and the nilradical of a ring R respectively. Also we denote the ring of integers modulo n by ${\mathbb{Z}}_n.$ Further K_n, $K_{m,n},$ P_n, and C_n denote respectively the complete graph on n vertices, complete bipartite graph with a bipartition into vertex sets of cardinality m and n, path on n vertices and cycle with n vertices. Moreover, the neighborhood set of a vertex v in G is $N_G(u)=\{v\in V(G):v\mathrm{ }\text{is}\mathrm{ }\text{adjacent}\mathrm{ }\text{to}\mathrm{ }u\text{in}G\}.$ All the definitions related to algebra are from Dummit and Foote [15], and the definitions related to graph theory are from Chartrand and Zhang [13].

2. Wiener index of graphs associated to rings

In recent years, there are numerous works done on computing or estimating the Wiener index of graphs associated with rings. Let us start with the zero-divisor graph.

In 1988, Beck [11] established the concept of a graph constructed with the zero-divisors of a commutative ring R. After altering Beck’s definition Anderson and Livingston [6] have developed the current concept, as well as the nomenclature for the zero-divisor graph in 1999. For more details on zero-divisor graphs, one may refer to [4]. The modified definition of the zero-divisor graph is as follows.

Definition 2.1.

[6] Let R be a commutative ring with identity and Z(R) be its set of zero-divisors. The zero-divisor graph of R, denoted by $\Gamma (R),$ is the simple graph with vertex set $Z{(R)}^{*}=Z(R)\setminus \left\{0\right\},$ and two distinct vertices x and y are adjacent if xy = 0.

Some of the initial works on computing Wiener index of zero-divisor graphs are based on algorithms. To begin with, the Wiener index of a zero-divisor graph of ${\mathbb{Z}}_n,$ where $n=p^2$ or pq, is determined by Ahmadi and Nezhad in [1]. The relevant results are stated below.

Theorem 2.2.

[1, Theorem 3] If p is a prime number, then $W(\Gamma ({\mathbb{Z}}_{p^2}))=\frac{(p-1)(p-2)}{2}.$

Theorem 2.3.

[1, Theorem 4] If p and q are two prime numbers, then $W(\Gamma ({\mathbb{Z}}_{pq}))=p^2+q^2+pq-4p-4q+5.$

The authors of [32] calculated the Wiener index of the zero-divisor graph of ${\mathbb{Z}}_{p^3}$ where p is a prime number.

Theorem 2.4.

[32, Theorem 5.1] If p is a prime number, then $W(\Gamma ({\mathbb{Z}}_{p^3}))=\frac{(p-1)}{2}(2p^3-3p-2).$

The following theorem is due to Pirzada et al. [29] which gives the Wiener index of $\Gamma ({\mathbb{Z}}_{p^{\alpha }})$ for all $\alpha \in \mathbb{N}$ and a prime p.

Theorem 2.5.

[29, Theorem 8] If α is a positive integer and p is a prime, then the Wiener index of $\Gamma ({\mathbb{Z}}_{p^{\alpha }})$ is given by $$\begin{array}{c}W(\Gamma ({\mathbb{Z}}_{p^{\alpha }}))=\sum _{i=1}^{\lceil \frac{\alpha }{2}\rceil -1}|V_i|(\left|V_i\right|-1)+\sum _{i=\lceil \frac{\alpha }{2}\rceil }^{\alpha -1}\frac{|V_i|(\left|V_i\right|-1)}{2}+2\sum _{i=1}^{\lceil \frac{\alpha }{2}\rceil -1}(\left|V_i\right|\sum _{j=i+1}^{\alpha -i-1}\left|V_j\right|)\\ +\sum _{r=1}^{\alpha -1}(|V_r|\stackrel{}{\underset{\alpha -j>r}{\sum _{j=1}^r}}|V_{\alpha -j}|)\end{array}$$ where $V_i=\{{\mathcal{l}}_i{p}^i:p\nmid {\mathcal{l}}_i\},1\le i\le \alpha -1$ is the partition of $V(\Gamma ({\mathbb{Z}}_{p^{\alpha }})).$

Recently, Asir and Rabikka [8] determined the Wiener index of the zero-divisor graph of $\Gamma ({\mathbb{Z}}_n)$ for all positive integers n. This will be presented as Theorem 2.6, Theorem 2.7 and Theorem 2.10. Here Theorem 2.6 which deals with the case where n is a prime power simplifies the calculation process of finding $W(\Gamma ({\mathbb{Z}}_{p^{\alpha }}))$ compared to the one given Theorem 2.5.

Theorem 2.6.

[8, Theorem 3.1] Let p be prime number and $\alpha \in {\mathbb{Z}}^{+}$. Then

1. $W(\Gamma ({\mathbb{Z}}_p))=W(\Gamma ({\mathbb{Z}}_4))=0.$

2. $W(\Gamma ({\mathbb{Z}}_{p^{\alpha }}))=\frac{1}{2}\left[2p^{2(\alpha -1)}-(\alpha -1)p^{\alpha }+(\alpha -6)p^{\alpha -1}+p^{\alpha -\lceil \frac{\alpha }{2}\rceil }+2\right]$ where $\alpha \ge 2$ and $p^{\alpha }\ne 4.$

Using Theorem 2.6, one can obtain Theorem 2.2 and Theorem 2.4. Next, we move on to remaining cases of positive integers. Actually, Theorem 2.7 and Theorem 2.10 provide a constructive method to calculate the value of Wiener index of zero-divisor graph of ${\mathbb{Z}}_n$ for any positive integer n.

Theorem 2.7.

[8, Theorem 3.4] Let $n=p_1\cdots p_k$ where p_i’s are distinct primes and $2\le k\in {\mathbb{Z}}^{+}$. Let d_j be a proper divisor of n for $j=1,\dots ,2^k-2.$

• For $1\le j\le k$, let $d_j=p_1\cdots p_{j-1}·p_{j+1}\cdots p_k$;

• For $k+1\le j\le 2^k-2$, let $d_j=p_1^{{\beta }_1}\cdots p_k^{{\beta }_k}$ where ${\beta }_i=0={\beta }_{i\prime }$ for some distinct $i,i\prime \in \{1,\dots ,k\}$. In this case, let $Z_j=\{i\in \{1,\dots ,k\}:{\beta }_i=0\}$ and let $Z_j=\{i_1,\dots ,i_{r(j)}\}$. For $1\le \mathcal{l}\le 2^{r(j)}-2$, define ${\tau }_{\mathcal{l}(j)}=p_{i_1}^{{\gamma }_{i_1}}\cdots p_{i_{r(j)}}^{{\gamma }_{i_{r(j)}}}$ where ${\gamma }_{i_s}\in \{0,1\}$ for all $1\le s\le r(j)$ with ${\gamma }_{i_t}=0$ for some $1\le t\le r(j).$

Then $$W(\Gamma ({\mathbb{Z}}_n))=\frac{1}{2}\left[\sum _{j=1}^{2^k-2}\left(\varphi \left(\frac{n}{d_j}\right)·(2(n-\varphi (n))-3-d_j)\right)\right]+\frac{1}{2}\left[\sum _{j=k+1}^{2^k-2}\left(\varphi \left(\frac{n}{d_j}\right)·\sum _{\mathcal{l}(j)=1}^{2^{r(j)}-2}\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(j)}}\right)\right)\right].$$

The following Corollary is an immediate consequence of Theorem 2.7.

Corollary 2.8.

Let p₁, p₂ and p₃ be distinct primes. Then

1. $W(\Gamma ({\mathbb{Z}}_{p_1{p}_2}))=p_1^2+p_2^2+p_1{p}_2-4p_1-4p_2+5.$

2. $W(\Gamma ({\mathbb{Z}}_{p_1{p}_2{p}_3}))=3(p_1^2{p}_2{p}_3+p_1{p}_2^2{p}_3+p_1{p}_2{p}_3^2)-15(p_1{p}_2{p}_3)+(p_1^2{p}_2^2+p_2^2{p}_3^2+p_1^2{p}_3^2)-3\left[p_1^2(p_2+p_3)+p_2^2(p_1+p_3)+p_3^2(p_1+p_2)\right]+8(p_1{p}_2+p_2{p}_3+p_1{p}_3)+2(p_1^2+p_2^2+p_3^2)-4(p_1+p_2+p_3)+3.$

The following remark explains the formula given in Theorem 2.7 with an example. The same example was considered in [8, Remark 3.6], however the details are more elaborated in the following remark.

Remark 2.9.

Consider $n=2·3·5·7=210.$ Then the number of proper divisors of n is $2^4-2=14.$ Here $d_1=3·5·7,d_2=2·5·7,d_3=2·3·7$ and $d_4=2·3·5.$ Let us take $d_5=2·3,d_6=2·5,d_7=2·7,d_8=3·5,d_9=3·7,d_{10}=5·7,d_{11}=2,d_{12}=3,d_{13}=5$ and $d_{14}=7.$

For instance, we illustrate d₁, d₆, d₁₁ and d₁₄. In general, let $d_j=2^{{\beta }_1}·3^{{\beta }_2}·5^{{\beta }_3}·7^{{\beta }_4}$ where ${\beta }_i\in \{0,1\}$ for $j=1,\dots ,14.$

Consider $d_1=3·5·7:$ Here ${\beta }_1=0.$ Implies that $Z_1=\left\{1\right\}$ and so $r(1)=1.$ Consequently ${\tau }_{1(1)}=2.$ Therefore $\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(1)}}\right)=1.$

Consider $d_6=2·5:$ Here ${\beta }_2=0$ and ${\beta }_4=0.$ Implies that $Z_6=\{2,4\}$ and so $r(6)=2.$ Consequently ${\tau }_{1(6)}=3$ and ${\tau }_{2(6)}=7.$ Therefore ${\sum }_{\mathcal{l}(6)=1}^2\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(6)}}\right)=24+8=32.$

For $d_{11}=2,$ we have $Z_{11}=\{2,3,4\}$ and so $r(11)=3.$ Consequently ${\tau }_{1(11)}=3,{\tau }_{2(11)}=5,{\tau }_{3(11)}=7,{\tau }_{4(11)}=3·5,{\tau }_{5(11)}=3·7$ and ${\tau }_{6(11)}=5·7.$ Therefore ${\sum }_{\mathcal{l}(11)=1}^6\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(11)}}\right)=24+12+8+6+4+2=56.$

For $d_{14}=7,$ we have $Z_{14}=\{1,2,3\}$ and so $r(14)=3.$ Consequently ${\tau }_{1(14)}=2,{\tau }_{2(14)}=3,{\tau }_{3(14)}=5,{\tau }_{4(14)}=2·3,{\tau }_{5(14)}=2·5$ and ${\tau }_{6(14)}=3·5.$ Therefore ${\sum }_{\mathcal{l}(14)=1}^6\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(14)}}\right)=48+24+12+24+12+6=126.$

Note that $n-\varphi (n)=210-48=160.$ $$\begin{array}{c}W(\Gamma ({\mathbb{Z}}_{210}))=\frac{1}{2}\left[\sum _{j=1}^{14}\left(\varphi \left(\frac{n}{d_j}\right)·(2\times 160-3-d_j)\right)\right]+\frac{1}{2}\left[\sum _{j=5}^{14}\left(\varphi \left(\frac{n}{d_j}\right)·\sum _{\mathcal{l}(j)=1}^{2^{r(j)}-2}\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(j)}}\right)\right)\right]\\ =\frac{1}{2}\left[50184+9120\right]=29652.\end{array}$$

The following theorem is the last part of finding the value of Wiener index of $\Gamma ({\mathbb{Z}}_n)$ for any $n\in {\mathbb{Z}}^{+}.$

Theorem 2.10.

[8, Theorem 3.8] Let $n=p_1^{{\alpha }_1}\cdots p_k^{{\alpha }_k}$, where p_i is a prime, ${\alpha }_i\in {\mathbb{Z}}^{+}$ with at least one ${\alpha }_i\ge 2$ and $2\le k\in {\mathbb{Z}}^{+}$. Rearrange p_i’s such that $n=p_1\cdots p_w·p_{w+1}^{{\alpha }_{w+1}}\cdots p_k^{{\alpha }_k}$ where ${\alpha }_i\ge 2$ for all $i=w+1,\dots ,k.$ $($In case of ${\alpha }_i\ge 2$ for all $i=1,\dots ,k$, take $w=0)$. Let $d_j=p_1^{{\beta }_1}\cdots p_k^{{\beta }_k}$ be a proper divisor of n for all $j=1,\dots ,({\prod }_{i=1}^k({\alpha }_i+1)-2)$. Arrange d_j’s in such a way that

• for $1\le j\le w,$ let $d_j=p_1\cdots p_{j-1}·p_{j+1}\cdots p_w·p_{w+1}^{{\alpha }_{w+1}}\cdots p_k^{{\alpha }_k}$;

• for $w+1\le j\le w+{\prod }_{i=1}^k\left(\lceil \frac{{\alpha }_i+1}{2}\rceil \right)-1,$ let ${\beta }_i\ge \lceil {\alpha }_i/2\rceil$ for all $i=1,\dots ,k$ and, for $w+{\prod }_{i=1}^k\lceil \frac{{\alpha }_i+1}{2}\rceil \le j\le w+{\prod }_{i=1}^k{\alpha }_i-1,$ let ${\beta }_i\ge 1$ and ${\beta }_{i\prime }<\lceil {\alpha }_{i\prime }/2\rceil$ for all $i=1,\dots ,k$ and some $i\prime \in \{1,\dots ,k\}$;

• for the remaining d_j’s, notate $j=w+{\prod }_{i=1}^k{\alpha }_i,\dots ,{\prod }_{i=1}^k({\alpha }_i+1)-2.$ In this case, let $Z_j=\{i\in \{1,\dots ,k\}:{\beta }_i=0\}$ and let $Z_j=\{i_1,\dots ,i_{r(j)}\}$. For $1\le \mathcal{l}\le {\sigma }_j$, define ${\tau }_{\mathcal{l}(j)}=p_{i_1}^{{\gamma }_{i_1}}\cdots p_{i_{r(j)}}^{{\gamma }_{i_{r(j)}}}$ where $0\le {\gamma }_{i_s}\le {\alpha }_{i_s}$ for all $1\le s\le r(j)$. In addition, if ${\beta }_m={\alpha }_m$ for all $m\in \{1,\dots ,k\}\setminus Z_j$, then there exists $t\in \{1,\dots ,r(j)\}$ such that ${\gamma }_{i_t}<{\alpha }_{i_t}.$

Then $$\begin{array}{c}W(\Gamma ({\mathbb{Z}}_n))=\frac{1}{2}[\sum _{j=1}^{({\prod }_{i=1}^k({\alpha }_i+1))-2}(\varphi (\frac{n}{d_j})·(2(n-\varphi (n))-3-d_j))]\\ +\frac{1}{2}[\sum _{j=w+{\prod }_{i=1}^k{\alpha }_i}^{({\prod }_{i=1}^k({\alpha }_i+1))-2}(\varphi (\frac{n}{d_j})·(\sum _{\mathcal{l}(j)=1}^{{\sigma }_j}\varphi (\frac{n}{{\tau }_{\mathcal{l}(j)}})))]+\frac{1}{2}(\sum _{j=w+1}^{({\prod }_{i=1}^k\lceil \frac{{\alpha }_i+1}{2}\rceil )-1}\varphi (\frac{n}{d_j})).\end{array}$$ where $${\sigma }_j=\{\begin{array}{cc}\left({\prod }_{s=1}^{r(j)}({\alpha }_{i_s}+1)\right)-2\hfill & \text{if}\mathrm{ }{\beta }_m={\alpha }_m\mathrm{ }\text{for}\mathrm{ }\text{all}\mathrm{ }m\in \{1,\dots ,k\}\setminus Z_j\hfill \\ \left({\prod }_{s=1}^{r(j)}({\alpha }_{i_s}+1)\right)-1\hfill & \text{otherwise}.\hfill \end{array}$$

Corollary 2.11.

Let p₁ and p₂ be distinct primes. Then $$W(\Gamma ({\mathbb{Z}}_{p_1{p}_2^2}))=\frac{1}{2}\left[6p_1{p}_2^3+2p_1^2{p}_2^2-12p_1{p}_2^2+2p_2^4-6p_2^3+3p_2^2+3p_2+2\right].$$

The following remark is an illustration of Theorem 2.10. The same example is listed in [8, Remark 3.10]. But Remark 3.10 [8] has some minor errors which was pointed out by the authors of [33] and that has been revised here. Also some additional calculation part is included in the following remark.

Remark 2.12.

Consider $n=5·7·2^3·3^2=2520.$ Here w = 2, ${\prod }_{i=1}^k\left(\lceil \frac{{\alpha }_i+1}{2}\rceil \right)-1=3,{\prod }_{i=1}^k{\alpha }_i-1=5$ and the number of proper divisors is ${\prod }_{i=1}^k({\alpha }_i+1)-2=46.$

• $d_1=7·2^3·3^2$ and $d_2=5·2^3·3^2.$

• Let $d_3=5·7·2^3·3, d_4=5·7·2^2·3, d_5=5·7·2^2·3^2$ and $d_6=5·7·2·3^2, d_7=5·7·2·3.$

• Let $d_8=5·7·2^3, d_9=5·7·2^2, d_{10}=5·7·2, d_{11}=5·7·3^2, d_{12}=5·7·3, d_{13}=5·2·3^2, d_{14}=5·2·3, d_{15}=5·2^2·3^2, d_{16}=5·2^2·3, d_{17}=5·2^3·3, d_8=7·2·3^2, d_{19}=7·2·3, d_{20}=7·2^2·3^2, d_{21}=7·2^2·3, d_{22}=7·2^3·3, d_{23}=5·7, d_{24}=5·2^3, d_{25}=5·2^2, d_{26}=5·2, d_{27}=5·3^2, d_{28}=5·3, d_{29}=7·2^3, d_{30}=7·2^2, d_{31}=7·2, d_{32}=7·3^2, d_{33}=7·3, d_{34}=2^3·3^2, d_{35}=2^3·3, d_{36}=2^2·3^2, d_{37}=2^2·3, d_{38}=2·3^2, d_{39}=2·3, d_{40}=5, d_{41}=7, d_{42}=2^3, d_{43}=2^2, d_{44}=2, d_{45}=3^2$ and $d_{46}=3.$

In general, take $d_j=5^{{\beta }_1}·7^{{\beta }_2}·2^{{\beta }_3}·3^{{\beta }_4}$ where ${\beta }_i\in \{1,\dots ,{\alpha }_i\}$ for $j=1,\dots ,46.$ For instance, we elaborate the terms in the formula of $W(\Gamma ({\mathbb{Z}}_n))$ for d₃₃ and d₄₁.

For $d_{12}=5·7·3$ since ${\beta }_3=0$ we have $Z_{12}=\left\{3\right\}.$ That is $r(12)=1,i_1=3.$ Here ${\beta }_4<{\alpha }_4,$ and so ${\sigma }_j=3.$ Consequently ${\tau }_{1(12)}=3^2, {\tau }_{2(12)}=2, {\tau }_{3(12)}=2^2.$ Therefore ${\sum }_{\mathcal{l}(12)=1}^3\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(12)}}\right)=96+288+144=528.$

For $d_{23}=5·7$ since ${\beta }_3={\beta }_4=0$ we have $Z_{24}=\{3,4\}.$ That is $r(23)=2,i_1=3$ and $i_2=4.$ Here ${\beta }_i={\alpha }_i$ for i = 1, 2, and so ${\sigma }_j=({\prod }_{s=1}^2({\alpha }_{i_s}+1))-2=10.$ Consequently ${\tau }_{1(23)}=3, {\tau }_{2(23)}=3^2, {\tau }_{3(23)}=2, {\tau }_{4(23)}=2^2, {\tau }_{5(23)}=2^3, {\tau }_{6(23)}=2·3, {\tau }_{7(23)}=2·3^2, {\tau }_{8(23)}=2^2·3, {\tau }_{9(23)}=2^2·3^2, {\tau }_{10(23)}=2^3·3.$ Therefore ${\sum }_{\mathcal{l}(23)=1}^{10}\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(23)}}\right)=192+96+288+144+144+96+48+48+24+48=1128.$

For $d_{33}=7·3;$ we have ${\beta }_1={\beta }_3=0,$ which implies that $Z_{33}=\{1,3\}.$ That is $r(33)=2,i_1=1$ and $i_2=3.$ Since ${\beta }_4<{\alpha }_4,$ we have ${\sigma }_j=({\prod }_{s=1}^2({\alpha }_{i_s}+1))-1=7.$ Consequently ${\tau }_{1(33)}=2, {\tau }_{2(33)}=2^2, {\tau }_{3(33)}=2^3, {\tau }_{4(33)}=5, {\tau }_{5(33)}=5·2, {\tau }_{6(33)}=5·2^2$ and ${\tau }_{7(33)}=5·2^3.$ Therefore ${\sum }_{\mathcal{l}(33)=1}^7\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(33)}}\right)=288+144+144+72+36+36+144=864.$

For $d_{41}=7;$ we have $Z_{41}=\{1,3,4\}.$ Since ${\beta }_2={\alpha }_2,{\sigma }_j=({\prod }_{s=1}^3({\alpha }_{i_s}+1))-2=22.$ Consequently, ${\tau }_{1(33)}=2, {\tau }_{2(33)}=2^2, {\tau }_{3(33)}=2^3, {\tau }_{4(33)}=3, {\tau }_{5(33)}=3^2, {\tau }_{6(33)}=5, {\tau }_{7(33)}=2·3, {\tau }_{8(33)}=2^2·3, {\tau }_{9(33)}=2^3·3, {\tau }_{10(33)}=2·3^2, {\tau }_{11(33)}=2^2·3^2, {\tau }_{12(33)}=2^3·3^2, {\tau }_{13(33)}=5·2, {\tau }_{14(33)}=5·2^2, {\tau }_{15(33)}=5·2^3, {\tau }_{16(33)}=5·3, {\tau }_{17(33)}=5·3^2, {\tau }_{18(33)}=5·2·3, {\tau }_{19(33)}=5·2^2·3, {\tau }_{20(33)}=5·2^3·3, {\tau }_{21(33)}=5·2·3^2$ and ${\tau }_{22(33)}=5·2^2·3^2.$ Therefore ${\sum }_{\mathcal{l}(41)=1}^{22}\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(11)}}\right)=288+144+144+192+96+144+96+48+48+48+24+24+72+36+36+48+24+24+12+12+12+6=1578.$

Note that $n-\varphi (n)=2520-576=1944.$ So $$\begin{array}{l}W(\Gamma ({\mathbb{Z}}_{2520}))=\frac{1}{2}\left[\sum _{j=1}^{46}\left(\varphi \left(\frac{n}{d_j}\right)·\left(2\times 1944-3-d_j\right)\right)\right]\\ +\frac{1}{2}\left[\sum _{j=8}^{46}\left(\varphi \left(\frac{n}{d_j}\right)·\left(\sum _{\mathcal{l}(j)=1}^{{\sigma }_j}\varphi \left(\frac{n}{{\tau }_{\mathcal{l}(j)}}\right)\right)\right)\right]+\frac{1}{2}\left(\sum _{j=3}^5\varphi \left(\frac{n}{d_j}\right)\right)\\ =\frac{1}{2}\left[7502511+1401984+5\right]=4452250.\end{array}$$

Recently, using Wiener matrix and composition of graphs Selvakumar et al. [33] derived a formula for the Wiener index of the zero-divisor graph of a finite commutative ring R. More specifically, they explicitly calculated the Wiener index of $\Gamma (R)$ when R is a reduced ring, ring of integers modulo n, and more generally the product of ring of integers modulo n. Before moving into those results, let us introduce the required definitions and notations.

Definition 2.13.

Let H be a graph with vertex set $V=\{1,\dots ,k\}$ and let $G_1,\dots ,G_k$ be a collection of graphs with the respective vertex sets $V_i=\{v_i^1,\dots ,v_i^{n_i}\}$ for $(1\le i\le k).$ Then their generalized composition $G=H[G_1,\dots ,G_k]$ has vertex set $V_1\bigsqcup \dots \bigsqcup V_k$ and two vertices $v_i^p$ and $v_j^p$ of $H[G_1,\dots ,G_k]$ are adjacent if one of the following condition is satisfied:

• $i=j,$ and $v_i^p$ and $v_j^q$ are adjacent vertices in G_i.

• $i\ne j$ and i and j are adjacent in H.

The graph H is said to be the base graph of G and the graphs G_i’s are called the factors of G.

Some more notions are defined in the following assumption.

Assumption 2.14.

Let R be an arbitrary finite commutative ring with unity. First, let us define an equivalence relation ∼ on $Z{(R)}^{*}$ as follows. For $x,y\in Z{(R)}^{*},$ define $x\sim y$ if and only if $\mathit{Ann}(x)=\mathit{Ann}(y)$ where $\mathit{Ann}(x)=\{a\in R:ax=0\}.$ Let $C_1,\dots ,C_k$ be the equivalence classes of this relation with respective representatives $c_1,\dots ,c_k.$ Then $Z{(R)}^{*}={\bigsqcup }_{i=1}^k{C}_i.$ Let us call these classes the equiv-annihilator classes of $\Gamma (R).$

Let G be a graph such that $G=H[G_1,\dots ,G_k]$ where H is a connected graph on k vertices and the factor graphs $G_i^{\prime }$s are either complete or empty (graph with vertices but no edges). Let $V(G_i)=\{v_i^1,\dots ,v_i^{n_i}\}$ be the vertex set of the graph G_i, then $V(G)={\bigsqcup }_{i=1}^{k}V(G_i).$ Let us define the Wiener matrix of the graph G as follows;

Definition 2.15.

The Wiener matrix $\mathcal{W}(G)$ of the graph G is defined as follows. The matrix $\mathcal{W}(G)$ is index by the vertex set of H. For two vertices $i,j\in H$, define $$\mathcal{W}{(G)}_{i,j}=\{\begin{array}{cc}\left(\begin{array}{c}\left|G_i\right|\\ 2\end{array}\right)\hfill & \mathrm{ }\text{if}\mathrm{ }i=j\mathrm{ }\text{and}\mathrm{ }{G}_i\mathrm{ }\text{is}\mathrm{ }\text{complete},\hfill \\ 2\left(\begin{array}{c}\left|G_i\right|\\ 2\end{array}\right)\hfill & \mathrm{ }\text{if}\mathrm{ }i=j\mathrm{ }\text{and}\mathrm{ }{G}_i\mathrm{ }\text{is}\mathrm{ }\text{empty},\hfill \\ \left|G_i\right|·\left|G_j\right|d_H(i,j)\hfill & \mathrm{ }\text{if}\mathrm{ }i\ne j\hfill \end{array}$$ where $\left|G_i\right|$ denotes the number of vertices in the factor graph G_i.

For the rest of the section, we will consider the notations from Definition 2.13, Assumption 2.14 and Definition 2.15 wherever required.

The following lemma proves that the graph $\Gamma (R)$ is a generalized composition of certain complete graphs and empty graphs. Using this observation the formula for the Wiener index of $\Gamma (R)$ is deduced.

Lemma 2.16.

[33, Proposition 1] Let C_i be as in Assumption 2.14. Let G_i be the subgraph induced by the set C_i in $\Gamma (R)$. Then $\Gamma (R)=H[G_1,\dots ,G_k].$

In [33], the authors have provided a formula for the Wiener index of the zero-divisor graph of an arbitrary finite commutative ring with unity. These approaches are different from [8], specifically they visualize the zero-divisor graph $\Gamma (R)$ as a generalized composition of suitable choices of graphs. The main formula for the Wiener index of zero-divisor graph of finite commutative ring is given in Theorem 2.19. Before moving into the main result, we need the two lemmas. The first lemma explains the distance between any two vertices of a graph G as defined in the Assumption 2.14.

Lemma 2.17.

[33, Lemma 3] Let $v_i^p$ and $v_j^q(1\le p\le n_i\mathrm{ }\text{and}\mathrm{ }1\le q\le n_j)$ be two arbitrary vertices of G. Then $$d_G(v_i^p,v_j^q)=\{\begin{array}{cc}1\hfill & \mathrm{ }\text{if}\mathrm{ }i=j\mathrm{ }\text{and}\mathrm{ }{G}_i\mathrm{ }\text{is}\mathrm{ }\text{complete}\mathrm{ }\hfill \\ 2\hfill & \mathrm{ }\text{if}\mathrm{ }i=j\mathrm{ }\text{and}\mathrm{ }{G}_i\mathrm{ }\text{is}\mathrm{ }\text{empty}\mathrm{ }\hfill \\ d_H(i,j)\hfill & \mathrm{ }\text{if}\mathrm{ }i\ne j\hfill \end{array}$$ where $d_G(v_i^p,v_j^q)$ denotes the distance between the vertices $v_i^p$ and $v_j^q$ in the graph G.

The second lemma deals with the Wiener index of the graph G which was given in Assumption 2.14.

Lemma 2.18.

[33, Lemma 4] Let G be the graph defined in Assumption 2.14. Then the Wiener index of G is given by $$W(G)=\sum _{1\le i\le k,G_i\mathrm{ }\text{is}\mathrm{ }\text{complete}\mathrm{ }}\left(\begin{array}{c}{n}_i\\ 2\end{array}\right)+\sum _{1\le i\le k,G_i\mathrm{ }\text{is}\mathrm{ }\text{empty}\mathrm{ }}2\left(\begin{array}{c}{n}_i\\ 2\end{array}\right)+\sum _{1\le i<j\le k}{n}_i{n}_j{d}_H(i,j).$$ where $\left|G_i\right|=n_i$. In particular, the Wiener index of G is equal to the total trace of the Wiener matrix $\mathcal{W}(G).$

Now we deduce a formula for the Wiener index of the zero-divisor graph of a commutative ring R. Note that, by Lemma 2.16, we have $\Gamma (R)=H[\Gamma (C_1),\dots ,\Gamma (C_k)]$ with each $\Gamma (C_i)$ either complete or empty. Hence, we can use Lemma 2.18 to calculate the Wiener index of $\Gamma (R).$

Theorem 2.19.

[33, Theorem 1] Let R be an arbitrary finite commutative ring with unity. Then the Wiener index of the graph $\Gamma (R)$ is equal to $$W(\Gamma (R))=\sum _{1\le i\le k,c_i^2=0}\left(\begin{array}{c}\left|C_i\right|\\ 2\end{array}\right)+\sum _{1\le i\le k,c_i^2\ne 0}2\left(\begin{array}{c}\left|C_i\right|\\ 2\end{array}\right)+\sum _{1\le i<j\le k}\left|C_i\right|·\left|C_j\right|d_H(c_i,c_j).$$

In particular, the Wiener index of $\Gamma (G)$ is equal to the total trace of the Wiener matrix $\mathcal{W}(\Gamma (G)).$

Further, note that the value of $d_H(c_i,c_j)$ can be deduced from Lemma 2.17.

The authors of [33] have also calculated the Wiener index of the zero-divisor graph $\Gamma (R)$ when R is a finite commutative reduced ring with unity. Before moving into the corresponding result, we need the following notations for the subsequent results.

Notation 2.20.

It is well-known that every finite commutative reduced ring with unity can be written as the product of finite fields. Let $R={\mathbb{F}}_{q_1}\times \cdots \times {\mathbb{F}}_{q_k}$ where $q_i=p_i^{k_i}$ and ${\mathbb{F}}_q$ is the finite field with q elements. We have $Z({\mathbb{F}}_q)=\left\{0\right\}.$ Let ${\mathcal{P}}_k=\{A\subseteq [k]:A\ne \varnothing$ and $A\ne [k]\}$ where $[k]:=\{1,\dots ,k\}.$ For $A\in {\mathcal{P}}_k,$ we define the characteristic vector of A to be the element ${\mathbf{1}}_A=(a_1,\dots ,a_k)\in R$ satisfying a_i = 1 if $i\in A$ and a_i = 0 otherwise. Also, for $A\in {\mathcal{P}}_k,$ define the sets $C_A=\{(a_1,\dots ,a_k)\in R:a_i\mathrm{ }\text{is}\mathrm{ }\text{non}-\text{zero}\text{if}\text{and}\text{only}\text{if}i\in A\}.$

Note that the vertex set of H is $V(H)=\{{\mathbf{1}}_A:A\in {\mathcal{P}}_k\}$ and the vertices ${\mathbf{1}}_A$ and ${\mathbf{1}}_B$ are adjacent in H if and only if ${\mathbf{1}}_A·{\mathbf{1}}_B=0$ if and only if $A\cap B=\varnothing .$

Lemma 2.21.

[33, Lemma 6] Let ${\mathbf{1}}_A$ and ${\mathbf{1}}_B$ be two distinct vertices of H then $$d_H({\mathbf{1}}_A,{\mathbf{1}}_B)=\{\begin{array}{cc}1\hfill & \mathit{ifA}\cap B=\varnothing ,\hfill \\ 2\hfill & \mathit{ifA}\cap B\ne \varnothing \mathrm{ }\text{and}\mathrm{ }A\cup B⊊\left[k\right],\hfill \\ 3\hfill & \mathit{ifA}\cap B\ne \varnothing \mathrm{ }\text{and}\mathrm{ }A\cup B=\left[k\right].\hfill \end{array}$$

Notation 2.22.

Define the sets $$\begin{array}{l}{D}_1=\left\{\right\{A,B\}:A,B\in {\mathcal{P}}_k,A\ne B\mathrm{ }\text{and}\mathrm{ }A\cap B=\varnothing \},\\ D_2=\left\{\right\{A,B\}:A,B\in {\mathcal{P}}_k,A\ne B,A\cap B\ne \varnothing \mathrm{ }\text{and}\mathrm{ }A\cup B⊊[k\left]\right\}\mathrm{ }\text{and}\mathrm{ }\\ D_3=\left\{\right\{A,B\}:A,B\in {\mathcal{P}}_k,A\ne B,A\cap B\ne \varnothing \mathrm{ }\text{and}\mathrm{ }A\cup B=[k\left]\right\}.\end{array}$$

The cardinality of the sets $D_1,D_2,$ and D₃ can be calculated using the following expression.

Proposition 2.23.

[33, Proposition 2] The sets D₁, D₂ and D₃ have the following cardinalities.

1. $\left|D_1\right|={\sum }_{i=2}^k(\begin{array}{c}k\\ i\end{array})(2^{i-1}-1)$

2. $\left|D_3\right|={\sum }_{i=1}^{k-1}(\begin{array}{c}k\\ i\end{array})(2^{i-1}-1)$

3. $\left|D_2\right|=\frac{1}{2}((\begin{array}{c}k\\ 2\end{array})\times (\begin{array}{c}k\\ 2\end{array})-(\begin{array}{c}k\\ 2\end{array}))-\left|D_1\right|-\left|D_3\right|$

The following result gives a expression for the Wiener index of zero-divisor graph of finite commutative reduced ring with unity. All the undefined notations in the following statement are from Notation 2.20 and Notation 2.22.

Theorem 2.24.

[33, Theorem 2] Let R be a finite commutative reduced ring with unity. Then the Wiener index of $\Gamma (R)$ is given by $$\begin{array}{c}W(\Gamma (R))=\sum _{A\in {\mathcal{P}}_k}2\left(\begin{array}{c}\prod _{i\in A}(q_i-1)\\ 2\end{array}\right)+\sum _{\{A,B\}\in D_1}(\prod _{i\in A}(q_i-1))(\prod _{i\in B}(q_i-1))\\ +\sum _{\{A,B\}\in D_2}2(\prod _{i\in A}(q_i-1))(\prod _{i\in B}(q_i-1))+\sum _{\{A,B\}\in D_3}3(\prod _{i\in A}(q_i-1))(\prod _{i\in B}(q_i-1)).\end{array}$$

Using Theorem 2.24, the authors of [33] have deduced Corollary 2.8 and Remark 2.9.

Example 2.25.

[33, Example 5] Assume that $R=\underset{\mathrm{k}-\text{times}}{\underbrace{{\mathbb{Z}}_2\times \cdots \times {\mathbb{Z}}_2}}.$ Then for each $A\in {\mathcal{P}}_k,$ we have $\left|C_A\right|=1.$ Therefore, the Wiener index of $\Gamma (R)$ can be written in the following form $W(\Gamma (R))=\left|D_1\right|+2\left|D_2\right|+3\left|D_3\right|.$ The values of $\left|D_i\right|$ for i = 1, 2, 3 can be calculated using Proposition 2.23.

Using Theorem 2.19, the authors of [33] have provided an expression for the Wiener index of the zero-divisor graph of product of ring of integers modulo n; Interested readers can refer [33, Theorem 4].

Moreover, Koam et al. [3] have determined several edge-based eccentric topological indices of a zero-divisor graph of some algebraic structures; Interested readers can refer [3] for corresponding results.

At the end of this section, we have mentioned some basic results regarding the Wiener index for line graph of zero-divisor graph of ${\mathbb{Z}}_n.$

Theorem 2.26.

[34, Theorem 3.5] If q is an odd prime number, then $W(L(\Gamma ({\mathbb{Z}}_{2q})))$ is $\frac{(q-1)(q-2)}{2}.$

Theorem 2.27.

[34, Theorem 3.6] If q > 3 is any prime number, then $W(L(\Gamma ({\mathbb{Z}}_{3q})))$ is $(q-1)(3q-5).$

Theorem 2.28.

[34, Theorem 3.7] If p, q are distinct odd prime numbers, then $W(L(\Gamma ({\mathbb{Z}}_{pq})))$ is $\frac{1}{2}(p-1)(q-1)(2pq-3p-3q+4).$

Theorem 2.29.

[34, Theorem 3.8] If p is any prime number and $p\ge 5$, then $W(L(\Gamma ({\mathbb{Z}}_{p^2})))$ is $\frac{1}{4}(p-1){(p-2)}^2(p-3).$

Let us close this section, by mentioning an open problem in this subject area.

Question 2.30. Find the Wiener index of $L(\Gamma ({\mathbb{Z}}_n)))$ for $n\ne p,p^2$ or pq where p and q are primes.

3. Wiener index of unit graphs and total graphs

In 1990, Grimaldi [16] discovered the unit graph for ${\mathbb{Z}}_n.$ For an arbitrary ring R, Ashrafi et al. [7] generalized the unit graph $G({\mathbb{Z}}_n)$ to G(R) in 2010. There are numerous other articles on this specific topic (see [10,23–26]). In [22], one can find a survey of the study on unit graphs.

Definition 3.1.

Let R be a commutative ring with nonzero identity and U(R) be the set of all units in R. The unit graph of R, denoted by G(R), has vertex set as the set of all elements of R, for distinct vertices x and y are adjacent if and only if $x+y\in U(R).$

In 2015, Pranjali and Acharya [30] proposed an algorithm to find the Wiener index of $G({\mathbb{Z}}_n)$ and using the algorithm they found the value of $W(G({\mathbb{Z}}_n))$ for $n=2,3,5,15,48.$ Recently, Asir et al. [9] have determined the Wiener index for unit graph of any finite commutative ring. Note that Theorem 3.2 and Theorem 3.3 gives the value of the Wiener index for unit graph of a finite commutative ring R in terms of cardinality of R and cardinality of U(R), the set of units of R.

Theorem 3.2.

[9, Theorem 2.5] Let R be a finite commutative ring. Then the following statements hold:

1. If R is isomorphic to a field with Char(R) = 2, then $$W(G(R))=\frac{|R{|}^2-|R|}{2}.$$

2. If R is isomorphic to one of the following rings

   1. R is a field with $\mathit{Char}(R)\ne 2$;

   2. R is not a field and R cannot have ${\mathbb{Z}}_2$ as a quotient;

   3. R is a local ring with maximal ideal $\mathfrak{m}$ such that $|R/\mathfrak{m}|=2$ and $R\ncong {\mathbb{Z}}_2,$

then $$W(G(R))=\{\begin{array}{cc}|R{|}^2-(\frac{|U(R)|}{2}+1)|R|,\hfill & \mathrm{ }\text{if}\mathrm{ }2\notin U(R)\hfill \\ |R{|}^2-(\frac{|U(R)|}{2}+1)|R|+\frac{|U(R)|}{2},\hfill & \mathrm{ }\text{if}\mathrm{ }2\in U(R)\hfill \end{array}$$

1. If R is isomorphic to a non-local ring with ${\mathbb{Z}}_2$ as a quotient and cannot have ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ as a quotient, then $$W(G(R))=\frac{5}{4}|R{|}^2-(|U(R)|+1)|R|.$$

Normally, Wiener index is meaningless for a disconnected graph. However, the authors of [21] have defined the Wiener index of a disconnected graph as the sum of all Wiener index of connected components of G. That is $W(G)=\sum _{i}W(G_i)$ where G_i is the connected components of G. Note that a component or block is a maximal connected subgraph of G.

Theorem 3.3.

[9, Theorem 2.6] If $R\mathrm{ }\text{has}\mathrm{ }{\mathbb{Z}}_2\times {\mathbb{Z}}_2$ as a quotient where $R\cong R_1\times \cdots \times R_p\times R_{p+1}\times \cdots \times R_n$ where each R_i is local with maximal ideal ${\mathfrak{m}}_i$ for $1\le i\le n$ and $R_j/{\mathfrak{m}}_j\cong {\mathbb{Z}}_2$ for $j=1,\dots ,p$ with $p\ge 2$, then $$W(G(R))=\{\begin{array}{cc}\frac{3}{2^{p+1}}|R{|}^2-|R|\hfill & \mathrm{ }\text{if}\mathrm{ }n=p\hfill \\ \frac{5}{2^{p+1}}|R{|}^2-(|U(R)|+1)|R|\hfill & \text{otherwise}\hfill \end{array}.$$

By applying the values of $\left|R\right|$ and $|U(R)|$ in Theorem 3.2, one will get the value of the Wiener index of any finite commutative ring. For instance, an example for this is given below. In what follows, the Euler phi-function for a positive integer n is denoted by $\varphi (n).$

Example 3.4.

[9, Example 2.7]

1. Let $R={\mathbb{Z}}_n.$

   1. If $n=2^m$ for some $m\in {\mathbb{Z}}^{+}$, then $W(G(R))=2^{2m}-2^{2m-2}-2^m.$

   2. If $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_n^{{\alpha }_k}$, where each p_i’s is an odd prime and ${\alpha }_i\in {\mathbb{Z}}^{+}$ for $i=1,\dots ,n$, then $W(G(R))=n^2-n-\frac{1}{2}(n-1)\varphi (n).$

   3. If $n=2^{{\alpha }_0}{p}_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_n^{{\alpha }_k}$, where each p_i’s is an odd prime and ${\alpha }_i\in {\mathbb{Z}}^{+}$ for $i=0,1,\dots ,n$, then $W(G(R))=\frac{5n^2}{4}-n(\varphi (n)+1).$

2. Let $R=\frac{{\mathbb{Z}}_n[x]}{〈x^2〉}.$

3. If $n=2^m$ for some $m\in {\mathbb{Z}}^{+}$, then $W(G(R))=2^{4m}-2^{4m-2}-2^{2m}.$

4. If $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\dots p_n^{{\alpha }_n}$ where each p_i’s is an odd prime and ${\alpha }_i\in {\mathbb{Z}}^{+}$ for $i=1,\dots ,n$, then $W(G(R))=n^4-n^2-\frac{n}{2}(n^2-1)\varphi (n).$

5. If $n=2^{{\alpha }_0}{p}_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\dots p_n^{{\alpha }_n}$ where each p_i’s is an odd prime and ${\alpha }_i\in {\mathbb{Z}}^{+}$ for $i=0,1,\dots ,n$, then $W(G(R))=\frac{5}{4}{n}^4-(n\varphi (n)+1)n^2.$

6. Let $R=\frac{{\mathbb{Z}}_n[x]}{〈x^2,xy,y^2〉}.$

   1. If $n=2^m$ for some $m\in {\mathbb{Z}}^{+}$, then $W(G(R))=2^{6m}-2^{6m-2}-2^{3m}.$

   2. If $n=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\dots p_n^{{\alpha }_n}$, where each p_i’s is an odd prime and ${\alpha }_i\in {\mathbb{Z}}^{+}$ for $i=1,\dots ,n$, then $W(G(R))=n^9-n^3-n^2(n^3-1)\varphi (n).$

   3. If $n=2^{{\alpha }_0}{p}_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\dots p_n^{{\alpha }_n}$, where each p_i’s is an odd prime and ${\alpha }_i\in {\mathbb{Z}}^{+}$ for $i=0,1,\dots ,n$, then $W(G(R))=\frac{5}{4}{n}^6-(n^2\varphi (n)+1)n^3.$

Next, let us see the result on hyper-Wiener index of the unit graph of a commutative ring. The following result determines the hyper-Wiener index for the graph G(R).

Theorem 3.5.

[9, Theorem 2.8] Let R be a finite commutative ring. Then the following statements hold:

1. If R is isomorphic to a field with Char(R) = 2, then $$HW(G(R))=|R{|}^2-|R|.$$

2. If R is isomorphic to one of the following rings

   1. R is a field with $\mathit{Char}(R)\ne 2$;

   2. R is not a field and R cannot have ${\mathbb{Z}}_2$ as a quotient;

   3. R is a local ring with maximal ideal $\mathfrak{m}$ such that $|R/\mathfrak{m}|=2$ and $R\ncong {\mathbb{Z}}_2,$

then $$HW(G(R))=\{\begin{array}{cc}3|R{|}^2-2|R|·|U(R)|-3|R|,\hfill & \mathrm{ }\text{if}\mathrm{ }2\notin U(R)\hfill \\ 3|R{|}^2-2|R|·|U(R)|-3|R|+2|U(R)|,\hfill & \mathrm{ }\text{if}\mathrm{ }2\in U(R)\hfill \end{array}$$

1. If R is isomorphic to a non-local ring with ${\mathbb{Z}}_2$ as a quotient and cannot have ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ as a quotient, then $$HW(G(R))=\frac{9}{2}|R{|}^2-5|R|·|U(R)|-3|R|.$$

In variation to the concept of zero-divisor graph, Anderson and Badawi [5] introduced the total graph of a commutative ring. There are numerous research articles which have been published on the total graph of commutative rings (see [2,5,27,36,37]).

Definition 3.6.

Let R be a commutative ring and Z(R) be the set of all zero-divisors of R. The total graph of R is a simple graph with all the elements of a ring as the vertices in which distinct $x,y\in R$ are adjacent if and only if $x+y\in Z(R).$

Let us close this section by mentioning some works on the Wiener index of total graph of a commutative ring. The first work in this direction provided the Wiener index of the total graph of ${\mathbb{Z}}_n$ when $n=p^2$ or pq for some primes p and q by Sheela and Om Prakash [35]. Here is the result on the Wiener index of total graphs.

Theorem 3.7.

[35, Theorem 4.1] If q is an odd prime, then $W(T_{\Gamma }({\mathbb{Z}}_{2q}))$ is $q^2+2q(q-1).$

Theorem 3.8.

[35, Theorem 4.2] If p, q are distinct odd primes, then $W(T_{\Gamma }({\mathbb{Z}}_{pq}))$ is $\frac{1}{2}(2p^2{q}^2-p^{2}q-p{q}^2-pq+p+q-1).$

Theorem 3.9.

[35, Theorem 4.3] If q is an odd prime, then $W(T_{\Gamma }({\mathbb{Z}}_{4q}))$ is $2q(6q-3)$

Theorem 3.10.

[35, Theorem 4.4] If p, q are distinct odd primes, then $W(T_{\Gamma }({\mathbb{Z}}_{p^{2}q}))=2p^4{q}^2-p^{4}q-p^3{q}^2+p^{3}q-2p^{2}q+p^2+pq-p.$

In 2014, Nikmehr et al. [28] found the Wiener index of total graph of ${\mathbb{Z}}_n$ for all the positive integer n. For $n=p^{\alpha },$ the total graph $T_{\Gamma }({\mathbb{Z}}_n)$ is disconnected. But, we can find the Wiener index of a disconnected graph using connected components of $T_{\Gamma }({\mathbb{Z}}_n).$ The relevant result is stated below.

Theorem 3.11.

Let p be a prime and $\alpha \in {\mathbb{Z}}^{+}$. Then the Wiener index of $T_{\Gamma }({\mathbb{Z}}_{p^{\alpha }})$ is as follows: $$W(T_{\Gamma }({\mathbb{Z}}_{p^{\alpha }}))=\{\begin{array}{cc}{2}^{\alpha -1}(2^{\alpha -1}-1)\hfill & \mathrm{ }\text{if}\mathrm{ }p=2,\hfill \\ \frac{3p^{2\alpha -1}-2p^{2\alpha -2}+p^{\alpha -1}-2p^{\alpha }}{2}\hfill & \text{otherwise}.\hfill \end{array}$$

Proof.

By [36, Theorem 3.2], we get that $$T_{\Gamma }({\mathbb{Z}}_{p^{\alpha }})=\{\begin{array}{cc}{K}_{2^{(\alpha -1)}}\cup K_{2^{(\alpha -1)}}\hfill & \text{if}\mathrm{ }p=2\hfill \\ K_{p^{(\alpha -1)}}\cup \underset{\frac{p-1}{2}\mathrm{ }\text{times}\mathrm{ }}{\underbrace{K_{p^{(\alpha -1)},p^{(\alpha -1)}}\cup K_{p^{(\alpha -1)},p^{(\alpha -1)}}\cup \dots \cup K_{p^{(\alpha -1)},p^{(\alpha -1)}}}}\hfill & \text{otherwise}.\hfill \end{array}$$

We know that the Wiener index of complete graph $K_{\mathcal{l}}$ is $\frac{\mathcal{l}(\mathcal{l}-1)}{2}$ and the Wiener index of complete bipartite graph $K_{\mathcal{l},\mathcal{l}}$ is $3{\mathcal{l}}^2-2\mathcal{l}.$ Therefore $W(T_{\Gamma }({\mathbb{Z}}_{2^{\alpha }}))=2^{\alpha -1}(2^{\alpha -1}-1)$ and $W(T_{\Gamma }({\mathbb{Z}}_{p^{\alpha }}))=\frac{p^{(\alpha -1)}(p^{(\alpha -1)}-1)}{2}+\frac{(p-1)}{2}(3p^{2(\alpha -1)}-2p^{(\alpha -1)})=\frac{1}{2}(3p^{2\alpha -1}-2p^{2\alpha -2}+p^{\alpha -1}-2p^{\alpha }).$ □

Theorem 3.12.

[28, Theorem 3.1] Let n be an integer and not a prime power. Then the Wiener index of $T_{\Gamma }({\mathbb{Z}}_n)$ is as follows: $$W(T_{\Gamma }({\mathbb{Z}}_n))=\{\begin{array}{cc}\frac{n(2n-m-1)}{2}\hfill & \mathrm{ }\text{if}\mathrm{ }n\mathrm{ }\text{is}\mathrm{ }\text{even},\hfill \\ \frac{(n-1)(2n-m)}{2}\hfill & \mathrm{ }\text{if}\mathrm{ }n\mathrm{ }\text{is}\mathrm{ }\text{odd}\hfill \end{array}$$

where $m=|Z({\mathbb{Z}}_n)|.$

The next result gives the value of the hyper-Wiener index of total graph of ${\mathbb{Z}}_n.$ First let us calculate the prime power case. Using a similar proof technique in Theorem 3.11, one can prove the following.

Theorem 3.13.

Let p be a prime and $\alpha \in {\mathbb{Z}}^{+}$. Then the hyper-Wiener index of $T_{\Gamma }({\mathbb{Z}}_{p^{\alpha }})$, is as follows: $$HW(T_{\Gamma }({\mathbb{Z}}_{p^{\alpha }}))=\{\begin{array}{cc}{2}^{\alpha }(2^{\alpha -1}-1)\hfill & \mathrm{ }\text{if}\mathrm{ }p=2,\hfill \\ 4p^{2\alpha -1}-3p^{2\alpha -2}+2p^{\alpha -1}-3p^{\alpha }\hfill & \text{otherwise}.\hfill \end{array}$$

Theorem 3.14.

[28, Theorem 3.2] Let n be an integer and not a prime power. Then the hyper-Wiener index of $\Gamma ({\mathbb{Z}}_n)$ is as follows: $$HW(T_{\Gamma }({\mathbb{Z}}_n))=\{\begin{array}{cc}\frac{n(3n-2m-1)}{2}\hfill & \mathrm{ }\text{if}\mathrm{ }n\mathrm{ }\text{is}\mathrm{ }\text{even},\hfill \\ \frac{(n-1)(3n-2m)}{2}\hfill & \mathrm{ }\text{if}\mathrm{ }n\mathrm{ }\text{is}\mathrm{ }\text{odd}\hfill \end{array}$$ where $m=|Z({\mathbb{Z}}_n)|.$

In the following example, we have listed the values for $W(T_{\Gamma }({\mathbb{Z}}_n))$ and $HW(T_{\Gamma }({\mathbb{Z}}_n))$ when $n=6,10,15,20.$

Example 3.15.

[28, Example 3.10] The values of $W(T_{\Gamma }({\mathbb{Z}}_n))$ and $HW(T_{\Gamma }({\mathbb{Z}}_n))$ for $n=6,10,15,20$ are as follows:

Table

n	$W(T_{\Gamma }({\mathbb{Z}}_n))$	$HW(T_{\Gamma }({\mathbb{Z}}_n))$
6	21	27
10	65	85
15	161	217
20	270	350

Let us close the section, by mentioning some open problems in the subject area.

Question 3.16. Let n be an integer and not a prime power. Then find the value of Wiener index of the total graph of ${\mathbb{Z}}_n$?

Question 3.17. Let R be a finite commutative ring. Find an expression for the Wiener index of total graph of R?

4. Wiener index of prime graphs

Another graph associated with ring is called a prime graph, which was introduced by Satyanarayana et al. [12] in 2010.

Definition 4.1.

Prime graph of a ring is defined as a graph whose vertices are all elements of the ring and any two distinct vertices $x,y\in R$ are adjacent if and only if $\mathit{xRy}=0\mathrm{ }\text{or}\mathrm{ }\mathit{yRx}=0.$ This graph is denoted by PG(R).

It is proved that if R is a semi-prime ring, then R is a prime ring if and only if the prime graph PG(R) is a tree. Recall that an ideal of a ring is said to be semi-prime if and only if it is an intersection of prime ideals of the ring and a semi-prime ring is one in which the zero ideal is semi-prime. For some specific cases of n, the authors in [19] have calculated the Wiener index of $PG({\mathbb{Z}}_n).$ The corresponding results are listed below.

Theorem 4.2.

[19, Theorem 3.2] If p is a prime, then $W(PG({\mathbb{Z}}_p))={(p-1)}^2.$

Theorem 4.3.

[19, Theorem 3.3] If p is a prime, then $W(PG({\mathbb{Z}}_{p^2}))=\frac{p(p-1)}{2}·[2p^2-2p+1].$

Theorem 4.4.

[19, Theorem 3.4] If p is a prime, then $W(PG({\mathbb{Z}}_{p^3}))=\frac{p(p-1)}{2}·[2p^4+2p^3-2p-3].$

The authors in [19] also evaluated the Wiener index of line graph of the prime graph $PG({\mathbb{Z}}_n).$ Note that the line graph $L(PG({\mathbb{Z}}_n))$ of the graph $PG({\mathbb{Z}}_n)$ is defined as the graph whose set of vertices constitutes of the edges of $PG({\mathbb{Z}}_n),$ where two vertices are adjacent if the corresponding edges have a common vertex in $PG({\mathbb{Z}}_n).$

Theorem 4.5.

[19, Theorem 4.19] Let n = 4 or an odd prime, then $W(L(PG({\mathbb{Z}}_n)))=\frac{n(n-1)}{2}.$

Let us close the section, by mentioning some open problems in the subject area.

Question 4.6. Find the Wiener index of $PG({\mathbb{Z}}_n)$ for $n\ne p^k,$ where p is prime and $k\in \{1,2,3\}.$

Question 4.7. Find the Wiener index of $L(PG({\mathbb{Z}}_n))$ for a non-prime integer n.

5. Conclusion

Even though the study of Wiener index of graphs from rings started in the year 2010, a decade of research in this area has brought out only some specific cases of graphs over ${\mathbb{Z}}_n.$ The three very recent works due to Asir [8], Selvakumar et al. [33] and Asir et al. [8] provided formulas for calculating the Wiener index of zero-divisor graphs of ${\mathbb{Z}}_n,$ zero-divisor graphs of any finite commutative ring with unity and unit graphs of any finite commutative ring with unity respectively. Now, these results opens the platform for finding the Wiener index of other well-studied classes of graphs from rings which includes annihilating-ideal graphs, annihilator graphs, comaximal graphs, Cayley graphs, Jacobson graphs, generalized total graphs, Cayley sum graphs and trace graphs of matrices.

The Wiener index is mostly used to determine the structure-property relationships. In particular, the Wiener index has a variety of applications in pharmaceutical science and in the structure of nanotubes. Further, the graphs constructed from ring structures are highly symmetric and so they have some remarkable properties in connecting chemical graph theory and networks in parallel computing. Thus, it is expected that the investigation process of finding Wiener index of graphs from rings may have some interesting applications in molecular graphs, theoretical computer science and networking.

Disclosure statement

No potential conflict of interest was reported by the authors.