On the cozero-divisor graphs associated to rings

Abstract

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by $\Gamma \prime (R),$ is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if $x\notin Ry$ and $y\notin Rx.$ In this paper, first we study the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_n).$ We show that the graph $\Gamma \prime ({\mathbb{Z}}_{pq})$ is Laplacian integral. Further, we obtain the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_n)$ for $n=p^{n_1}{q}^{n_2},$ where $n_1,n_2\in \mathbb{N}$ and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of $\Gamma \prime ({\mathbb{Z}}_n),$ we characterized the values of n for which the Laplacian spectral radius is equal to the order of $\Gamma \prime ({\mathbb{Z}}_n).$ Moreover, the values of n for which the algebraic connectivity and vertex connectivity of $\Gamma \prime ({\mathbb{Z}}_n)$ coincide are also described. At the final part of this paper, we obtain the Wiener index of $\Gamma \prime ({\mathbb{Z}}_n)$ for arbitrary n.

Keywords:

• Cozero-divisor graph
• ring of integers modulo n
• Laplacian spectrum
• Wiener index

Mathematics Subject Classification:

• 05C25
• 05C50

1. Introduction

The study of algebraic structures through graph theoretic properties has emerged as a fascinating research discipline in the past three decades, it has provided not only intriguing and exciting results but also opened up a whole new domain yet to be explored. At the beginning, the idea to associate a graph with ring structure was appeared in [9]. The cozero-divisor graph related to a commutative ring was introduced by Afkhami et al. in [1]. The cozero-divisor graph of a ring R with unity, denoted by $\Gamma \prime (R),$ is an undirected simple graph whose vertex set is the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if $x\notin Ry$ and $y\notin Rx.$ They discussed certain basic properties on the structure of cozero-divisor graph and studied the relationship between the zero divisor graph and the cozero-divisor graph over ring structure. In [2], they investigated the complement of cozero-divisor graph and characterized the commutative rings with forest, star or unicyclic cozero-divisor graphs. Akbari et al. [5], studied the cozero-divisor graph associated to the polynomial ring. Some of the work associated with the cozero-divisor graph on the rings can be found in [3, 4, 6, 8, 17, 18]. The spectral graph theory is associated with spectral properties including investigation of charateristic polynomials, eigenvalues, eiegnvectors of matrices related with graphs. Recently, Chattopadhyay et al. [11] studied the Laplacian spectrum of the zero divisor graph of the ring ${\mathbb{Z}}_n.$ They proved that the zero divisor graph of the ring ${\mathbb{Z}}_{p^l}$ is Laplacian integral for every prime p and a positive integer $l\ge 2.$ The work on spectral radius, viz. adjacency spectrum, Laplacian spectrum, signless Laplacian spectrum, distance signless spectrum etc., of the zero-divisor graphs can be found in [11, 16, 19, 20, 21, 22, 23]. The Wiener index, which is a distance based topological index, has various applications in pharmaceutical science, chemistry etc., see [12, 14, 26, 27]. Recently, the Wiener index of the zero divisor graph of the ring ${\mathbb{Z}}_n$ of integers modulo n has been studied in [7].

In this paper, we study the Laplacian spectrum and the Wiener index of the cozero-divisor graph associated with the ring ${\mathbb{Z}}_n.$ The paper is arranged as follows: In Sec. 2, we recall necessary results and fix our notations which are used throughout the paper. In Sec. 3, we study the structure of $\Gamma \prime ({\mathbb{Z}}_n).$ Section 4 deals with the Laplacian spectrum of the cozero-divisor graph of the ring ${\mathbb{Z}}_n$ for $n=p^{n_1}{q}^{n_2},$ where p, q are distinct primes. In Sec. 5, the Laplacian spectral radius and the algebraic connectivity of $\Gamma \prime ({\mathbb{Z}}_n)$ have been investigated. The Wiener index of $\Gamma \prime ({\mathbb{Z}}_n)$ has been obtained in Sec. 6.

2. Preliminaries

In this section, we recall necessary definitions, results and notations of graph theory from [25]. A graph Γ is a pair $\Gamma =(V,E),$ where $V=V(\Gamma )$ and $E=E(\Gamma )$ are the set of vertices and edges of Γ, respectively. Let Γ be a graph. The order of a graph Γ is the number of vertices of Γ. Two distinct vertices $x,y\in \Gamma$ are $\mathit{adjacent},$ denoted by $x\sim y,$ if there is an edge between x and y. Otherwise, we denote it by $x\nsim y.$ The set $N_{\Gamma }(x)$ of all the vertices adjacent to x in Γ is said to be the neighbourhood of x. A subgraph $\Gamma \prime$ of a graph Γ is a graph such that $V(\Gamma \prime )\subseteq V(\Gamma )$ and $E(\Gamma \prime )\subseteq E(\Gamma ).$ If $U\subseteq V(\Gamma )$ then the subgraph of Γ induced by U, denoted by $\Gamma (U),$ is the graph with vertex set U and two vertices of $\Gamma (U)$ are adjacent if and only if they are adjacent in Γ. The complement $\overline{\Gamma }$ of Γ is a graph with same vertex set as Γ and distinct vertices x, y are adjacent in $\overline{\Gamma }$ if they are not adjacent in Γ. A graph Γ is said to be complete if every two distinct vertices are adjacent. The complete graph on n vertices is denoted by K_n. A path in a graph is a sequence of distinct vertices with the property that each vertex in the sequence is adjacent to the next vertex of it. The graph Γ is said to be connected if there is path between every pair of vertex.

The distance between any two vertices x and y of Γ, denoted by d(x, y), is the number of edges in a shortest path between x and y. The Wiener index is defined as the sum of all distances between every pair of vertices in the graph that is the Wiener index of a graph Γ is given by $$W(\Gamma )=\frac{1}{2}\sum _{u\in V(\Gamma )}\sum _{v\in V(\Gamma )}d(u,v)$$ The diameter of a connected graph Γ, written as diam$(\Gamma ),$ is the maximum of the distances between vertices. If the graph consists of a single vertex, then the diameter is 0. The degree of a vertex $v\in \Gamma ,$ denoted by $\text{deg}(v),$ is the number of edges adjacent to v. The smallest degree among the vertices of Γ is called the minimum degree of Γ and it is denoted by $\delta (\Gamma ).$ A vertex cut-set in a connected graph Γ is a set X of vertices such that the remaining subgraph $\Gamma \setminus X,$ by removing the set X is disconnected or has only one vertex. The vertex connectivity of a connected graph Γ, denoted by $\kappa (\Gamma ),$ is the minimum size of a vertex cut set. Let ${\Gamma }_1$ and ${\Gamma }_2$ be two graphs. The union ${\Gamma }_1\cup {\Gamma }_2$ is the graph with $V({\Gamma }_1\cup {\Gamma }_2)=V({\Gamma }_1)\cup V({\Gamma }_2)$ and $E({\Gamma }_1\cup {\Gamma }_2)=E({\Gamma }_1)\cup E({\Gamma }_2).$ The join ${\Gamma }_1\vee {\Gamma }_2$ of ${\Gamma }_1$ and ${\Gamma }_2$ is the graph obtained from the union of ${\Gamma }_1$ and ${\Gamma }_2$ by adding new edges from each vertex of ${\Gamma }_1$ to every vertex of ${\Gamma }_2.$

Let Γ be a graph on k vertices and $V(\Gamma )=\{u_1,u_2,\dots ,u_k\}.$ Suppose that ${\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_k$ are k pairwise disjoint graphs. Then the generalised join graph $\Gamma [{\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_k]$ of ${\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_k$ is the graph formed by replacing each vertex u_i of Γ by Γ_i and then joining each vertex of Γ_i to every vertex of Γ_j whenever $u_i\sim u_j$ in Γ (cf. [24]).

For a finite simple (without multiple edge and loops) undirected graph Γ with vertex set $V(\Gamma )=\{u_1,u_2,\dots ,u_k\},$ the adjacency matrix $A(\Gamma )$ is defined as the k × k  matrix whose $(i,j)th$ entry is 1 if $u_i\sim u_j$ and 0 otherwise. We denote the diagonal matrix by $D(\Gamma )=\text{diag}(d_1,d_2,\dots ,d_k),$ where d_i is the degree of the vertex u_i of Γ. The Laplacian matrix $\mathcal{L}(\Gamma )$ of Γ is the matrix $D(\Gamma )-A(\Gamma ).$ The matrix $\mathcal{L}(\Gamma )$ is a symmetric and positive semidefinite, so that its eigenvalues are real and non-negative. Furthermore, the sum of each row (column) of $\mathcal{L}(\Gamma )$ is zero. The eigenvalues of $\mathcal{L}(\Gamma )$ are called the Laplacian eigenvalues of Γ and are taken as ${\lambda }_1(\Gamma )\ge {\lambda }_2(\Gamma )\ge \cdots \ge {\lambda }_n(\Gamma )=0.$ The second smallest Laplacian eigenvalue of $\mathcal{L}(\Gamma ),$ denoted by $\mu (\Gamma ),$ is called the algebraic connectivity of Γ. The largest Laplacian eigenvalue $\lambda (\Gamma )$ of $\mathcal{L}(\Gamma )$ is called the Laplacian spectral radius of Γ. Now let ${\lambda }_1(\Gamma )\ge {\lambda }_2(\Gamma )\ge \cdots \ge {\lambda }_r(\Gamma )=0$ be the distinct eigenvalues of Γ with multiplicities ${\mu }_1,{\mu }_2,\dots ,{\mu }_r,$ respectively. The Laplacian spectrum of Γ, that is the spectrum of $\mathcal{L}(\Gamma ),$ is represented as $$\Phi (\mathcal{L}(\Gamma ))=\left(\begin{array}{cccc}{\lambda }_1(\Gamma )& {\lambda }_2(\Gamma )& \cdots & {\lambda }_r(\Gamma )\\ {\mu }_1& {\mu }_2& \cdots & {\mu }_r\end{array}\right).$$ Sometime we write $\Phi (\mathcal{L}(\Gamma )$ as ${\Phi }_{\mathcal{L}}(\Gamma )$ also. The following results are useful in the sequel.

Theorem 2.1.

[10] Let Γ be a graph on k vertices having $V(\Gamma )=\{u_1,u_2,\dots ,u_k\}$ and let ${\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_k$ be k pairwise disjoint graphs on $n_1,n_2,\dots ,n_k$ vertices, respectively. Then the Laplacian spectrum of $\Gamma [{\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_k]$ is given by (1) $${\Phi }_L(\Gamma [{\Gamma }_1,{\Gamma }_2,\dots ,{\Gamma }_k])=\underset{i=1}{\overset{k}{\cup }}(D_i+({\Phi }_L({\Gamma }_i)\setminus \left\{0\right\}))\cup \Phi (\mathbb{L}(\Gamma ))$$(1) where $$D_i=\{\begin{array}{cc}\sum _{u_j\sim u_i}{n}_j\hfill & \text{if} N_{\Gamma }(u_i)\ne \varnothing ;\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$ (2) $$\mathbb{L}(\Gamma )=\left[\begin{array}{cccc}{D}_1& -p_{1,2}& \cdots & -p_{1,k}\\ -p_{2,1}& D_2& \cdots & -p_{2,k}\\ \cdots & \cdots & \cdots & \cdots \\ -p_{k,1}& -p_{k,2}& \cdots & D_k\end{array}\right]$$(2) such that $$p_{i,j}=\{\begin{array}{cc}\sqrt{n_i{n}_j}\hfill & if u_i\sim u_j in \Gamma \hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$ in (1), $({\Phi }_L({\Gamma }_i)\setminus \{0\}))$ means that one copy of the eigenvalue 0 is removed from the multiset ${\Phi }_L({\Gamma }_i)$, and $D_i+({\Phi }_L({\Gamma }_i)\setminus \{0\}))$ means D_i is added to each element of $({\Phi }_L({\Gamma }_i)\setminus \{0\})).$

Let Γ be a weighted graph by assigning the weight $n_i=|V({\Gamma }_i)|$ to the vertex u_i of Γ and i varies from 1 to k. Consider $L(\Gamma )=(l_{i,j})$ to be a k × k matrix, where $$l_{i,j}=\{\begin{array}{cc}-n_j\hfill & if i\ne j \mathit{and} u_i\sim u_j;\hfill \\ \sum _{u_i\sim u_r}{n}_r\hfill & if i=j;\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

The matrix $L(\Gamma )$ is called the vertex weighted Laplacian matrix of Γ, which is a zero row sum matrix but not a symmetric matrix in general. Though the k × k  matrix $\mathbb{L}(\Gamma )$ defined in Theorem 2.1, is a symmetric matrix but it need not be a zero row sum matrix. Since the matrices $\mathbb{L}(\Gamma )$ and $L(\Gamma )$ are similar, we have the following remark.

Remark 2.2.

$\Phi (\mathbb{L}(\Gamma ))=\Phi (L(\Gamma )).$

Let ${\mathbb{Z}}_n$ denotes the ring of integers modulo n that is, ${\mathbb{Z}}_n=\{0,1,\dots ,n-1\}.$ The number of integers which are prime to n and less than n is denoted by Euler Totient function $\varphi (n).$ An integer d, where $1<d<n,$ is called a proper divisor of n if $d|n.$ If d does not divide n then we write it as $d\nmid n.$ The number of all the divisors of n is denoted by $\tau (n).$ The greatest common divisor of the two positive integers a and b is denoted by gcd(a, b). The ideal generated by the element a of ${\mathbb{Z}}_n$ is the set $\{xa:x\in {\mathbb{Z}}_n\}$ and it is denoted by $〈a〉.$

3. Structure of the cozero-divisor graph $\Gamma \prime ({\mathbb{Z}}_n)$

In this section, we discuss about the structure of the cozero-divisor graph $\Gamma \prime ({\mathbb{Z}}_n).$ Let $d_1,d_2,\dots ,d_k$ be the proper divisors of n. For $1\le i\le k,$ consider the following sets $${\mathcal{A}}_{d_i}=\{x\in {\mathbb{Z}}_n:\text{gcd}(x,n)=d_i\}.$$

Remark 3.1.

For $x,y\in {\mathcal{A}}_{d_i},$ we have $〈x〉=〈y〉=〈d_i〉.$ Further, note that the sets ${\mathcal{A}}_{d_1},{\mathcal{A}}_{d_2},\dots ,{\mathcal{A}}_{d_k}$ forms a partition of the vertex set of the graph $\Gamma \prime ({\mathbb{Z}}_n).$ Thus, $V(\Gamma \prime ({\mathbb{Z}}_n))={\mathcal{A}}_{d_1}\cup {\mathcal{A}}_{d_2}\cup \cdots \cup {\mathcal{A}}_{d_k}.$

The cardinality of each ${\mathcal{A}}_{d_i}$ is known in the following lemma.

Lemma 3.2.

[28] $\left|{\mathcal{A}}_{d_i}\right|=\varphi \left(\frac{n}{d_i}\right)$ for $1\le i\le k.$

Lemma 3.3.

Let $x\in {\mathcal{A}}_{d_i},y\in {\mathcal{A}}_{d_j}$, where $i,j\in \{1,2,\dots ,\tau (n)-2\}$. Then $x\sim y$ in $\Gamma \prime ({\mathbb{Z}}_n)$ if and only if $d_i\nmid d_j$ and $d_j\nmid d_i.$

Proof.

First note that in ${\mathbb{Z}}_n,x\in 〈y〉$ if and only if $y|x.$ Let $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}$ be two distinct vertices of $\Gamma \prime ({\mathbb{Z}}_n).$ Suppose that $x\sim y$ in $\Gamma \prime ({\mathbb{Z}}_n).$ Then $x\notin 〈y〉$ and $y\notin 〈x〉.$ If $d_i|d_j,$ then $d_j\in 〈d_i〉=〈x〉.$ It follows that $〈y〉=〈d_j〉\subseteq 〈x〉$ and so $y\in 〈x〉,$ which is not possible. Similarly, if $d_j|d_i,$ then we get $x\in 〈y〉,$ again a contradiction. Thus, neither $d_i|d_j$ nor $d_j|d_i.$ Conversely, if $d_i\nmid d_j$ and $d_j\nmid d_i$ then we obtain $x\notin 〈y〉$ and $y\notin 〈x〉.$ It follows that $x\sim y.$ The result holds. □

For distinct vertices x, y of ${\mathcal{A}}_{d_i},$ by Remark 3.1, clearly $x\in 〈y〉$ and $y\in 〈x〉.$ It follows that $x\nsim y$ in $\Gamma \prime ({\mathbb{Z}}_n).$ Using Lemma 3.2, we have the following corollary.

Corollary 3.4.

The following statements hold:

1. For $i\in \{1,2,\dots ,\tau (n)-2\}$, the induced subgraph $\Gamma \prime ({\mathcal{A}}_{d_i})$ of $\Gamma \prime ({\mathbb{Z}}_n)$ is isomorphic to ${\overline{K}}_{\varphi \left(\frac{n}{d_i}\right)}.$

2. For $i,j\in \{1,2,\dots ,\tau (n)-2\}$ and $i\ne j$, a vertex of ${\mathcal{A}}_{d_i}$ is adjacent to either all or none of the vertices of ${\mathcal{A}}_{d_j}.$

Thus, the partition ${\mathcal{A}}_{d_1},{\mathcal{A}}_{d_2},\dots ,{\mathcal{A}}_{d_{\tau (n)-2}}$ of $V(\Gamma \prime ({\mathbb{Z}}_n))$ is an equitable partition in such a way that every vertex of the ${\mathcal{A}}_{d_i}$ has equal number of neighbors in ${\mathcal{A}}_{d_j}$ for every $i,j\in \{1,2,\dots ,\tau (n)-2\}.$

We define $\Upsilon {\prime }_n$ by the simple undirected graph whose vertex set is the set of all proper divisors $d_1,d_2,\dots ,d_k$ of n and two distinct vertices d_i and d_j are adjacent if and only if $d_i\nmid d_j$ and $d_j\nmid d_i.$

Lemma 3.5.

For a prime p, the graph $\Upsilon {\prime }_n$ is connected if and only if $n\ne p^t$, where $t\ge 3.$

Proof.

Suppose that $\Upsilon {\prime }_n$ is a connected graph and $V(\Upsilon {\prime }_n)=\{d_1,d_2,\dots ,d_k\}.$ If $n=p^t$ for $t\ge 3,$ then $V(\Upsilon {\prime }_{p^t})=\{p,p^2,\dots ,p^{t-1}\}.$ The definition of $\Upsilon {\prime }_n$ gives that $\Upsilon {\prime }_{p^t}$ is a null graph on t – 1 vertices. Thus, $\Upsilon {\prime }_n$ is not connected; a contradiction. Conversely, suppose that $n\ne p^t,$ where $t\ge 3.$ If $n=p^t$ for $t\in \{1,2\},$ then there is nothing to prove because $\Upsilon {\prime }_p$ is an empty graph whereas $\Upsilon {\prime }_{p^2}$ is a graph with one vertex only. We may now suppose that $n=p_1^{n_1}{p}_2^{n_2}\cdots p_m^{n_m},$ where p_i’s are distinct primes and $m\ge 2.$ Now let $d,d\prime \in V(\Upsilon {\prime }_n).$ If $d\nmid d\prime$ and $d\prime \nmid d,$ then $d\sim d\prime .$ Without loss of generality, assume that $d|d\prime$ with $d=p_1^{{\beta }_1}{p}_2^{{\beta }_2}\cdots p_m^{{\beta }_m}$ and $d\prime =p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_m^{{\alpha }_m}.$ Note that ${\alpha }_i,{\beta }_i\in \mathbb{N}\cup \left\{0\right\}$ such that ${\beta }_i\le {\alpha }_i.$ Since $d\prime$ is a proper divisor of n there exists $r\in \{1,2,\dots ,m\},$ where ${\alpha }_r\le n_r,$ such that $p_r^{n_r}\nmid d\prime$ and $d\prime \nmid p_r^{n_r}.$ Clearly, $p_r^{n_r}\nmid d.$ If $d\nmid p_r^{n_r},$ then $d\sim p_r^{n_r}\sim d\prime .$ If $d|p_r^{n_r},$ then there exists $s\in \{1,2,\dots ,m\}\setminus \left\{r\right\}$ such that $d\nmid p_s$ and $p_s\nmid d.$ Also, $p_r^{n_r}\nmid p_s$ and $p_s\nmid p_r^{n_r}.$ It follows that $d\prime \sim p_r^{n_r}\sim p_s\sim d.$ Hence, the graph $\Upsilon {\prime }_n$ is connected. □

Lemma 3.6.

$\Gamma \prime ({\mathbb{Z}}_n)=\Upsilon {\prime }_n[\Gamma \prime ({\mathcal{A}}_{d_1}),\Gamma \prime ({\mathcal{A}}_{d_2}),\dots ,\Gamma \prime ({\mathcal{A}}_{d_k})],$ where $d_1,d_2,\dots ,d_k$ are all the proper divisors of n.

Proof.

Replace the vertex d_i of $\Upsilon {\prime }_n$ by $\Gamma \prime ({\mathcal{A}}_{d_i})$ for $1\le i\le k.$ Consequently, the result can be obtained by using Lemma 3.3. □

Lemma 3.7.

For a prime p, we have $\Gamma \prime ({\mathbb{Z}}_n)$ is connected if and only if either n = 4 or $n\ne p^t$, where $t\ge 2.$

Proof.

Suppose that $\Gamma \prime ({\mathbb{Z}}_n)$ is a connected graph and $n\ne 4.$ If possible, let $n=p^t$ for $t\ge 2$ then note that $V(\Gamma \prime ({\mathbb{Z}}_n))=\Gamma \prime ({\mathcal{A}}_p)\cup \Gamma \prime ({\mathcal{A}}_{p^2})\cup \cdots \cup \Gamma \prime ({\mathcal{A}}_{p^{t-1}})$ and so $x\nsim y$ for any $x,y\in V(\Gamma \prime ({\mathbb{Z}}_n))$ (see Lemma 3.3 and Corollary 3.4). Consequently, $\Gamma \prime ({\mathbb{Z}}_n)$ is a null graph; a contradiction. Thus, $n\ne p^t,$ where $t\ge 2.$ Converse follows by the proof of Lemma 3.5 and Lemma 3.6. □

Example 3.8.

The cozero-divisor graph $\Gamma \prime ({\mathbb{Z}}_{30})$ is shown in Figure 1.

By Lemma 3.6, note that $\Gamma \prime ({\mathbb{Z}}_{30})=\Upsilon {\prime }_{30}[\Gamma \prime ({\mathcal{A}}_2),\Gamma \prime ({\mathcal{A}}_3),\Gamma \prime ({\mathcal{A}}_5),\Gamma \prime ({\mathcal{A}}_6),\Gamma \prime ({\mathcal{A}}_{10}),\Gamma \prime ({\mathcal{A}}_{15})],$ where $\Upsilon {\prime }_{30}$ is shown in Figure 2 and $\Gamma \prime ({\mathcal{A}}_2)={\overline{K}}_8,\Gamma \prime ({\mathcal{A}}_3)={\overline{K}}_4=\Gamma \prime ({\mathcal{A}}_6),\Gamma \prime ({\mathcal{A}}_5)={\overline{K}}_2=\Gamma \prime ({\mathcal{A}}_{10}),\Gamma \prime ({\mathcal{A}}_{15})={\overline{K}}_1.$

Figure 1. The graph $\Gamma \prime ({\mathbb{Z}}_{30}).$

Figure 2. The graph $\Upsilon {\prime }_{3}0.$

4. Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_n)$

In this section, we investigate the Laplacian spectrum of the $\Gamma \prime ({\mathbb{Z}}_n)$ for various n. Consider $d_1,d_2,\dots ,d_k$ as all the proper divisors of n. For $1\le i\le k,$ we give the weight $\varphi \left(\frac{n}{d_i}\right)=\left|{\mathcal{A}}_{d_i}\right|$ to the vertex d_i of the graph $\Upsilon {\prime }_n.$ Define the integer $$D_{d_j}=\sum _{d_i\in N_{\Upsilon {\prime }_n}(d_j)}\varphi \left(\frac{n}{d_i}\right)$$ The k × k  weighted Laplacian matrix $L(\Upsilon {\prime }_n)$ of $\Upsilon {\prime }_n$ defined in Theorem 2.1 is given by (3) $$L(\Upsilon {\prime }_n)=\left[\begin{array}{cccc}{D}_{d_1}& -l_{1,2}& \cdots & -l_{1,k}\\ -l_{2,1}& D_{d_2}& \cdots & -l_{2,k}\\ \cdots & \cdots & \cdots & \cdots \\ -l_{k,1}& -l_{k,2}& \cdots & D_{d_k}\end{array}\right]$$(3) where $$l_{i,j}=\{\begin{array}{cc}\varphi \left(\frac{n}{d_j}\right)\hfill & \text{if} d_i\sim d_j \text{in} \Upsilon {\prime }_n;\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$$

Theorem 4.1.

The Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_n)$ is given by ${\Phi }_L(\Gamma \prime ({\mathbb{Z}}_n))={\cup }_{i=1}^k(D_{d_i}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{d_i}))\setminus \left\{0\right\}))\cup \Phi (L(\Upsilon {\prime }_n)),$ where $D_{d_i}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{d_i}))\setminus \{0\})$ represents that $D_{d_i}$ is added to each element of the multiset $({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{d_i}))\setminus \{0\}).$

Proof.

By Lemma 3.6, $\Gamma \prime ({\mathbb{Z}}_n)=\Upsilon {\prime }_n[\Gamma \prime ({\mathcal{A}}_{d_1}),\Gamma \prime ({\mathcal{A}}_{d_2}),\dots ,\Gamma \prime ({\mathcal{A}}_{d_k})].$ Consequently, by Theorem 2.1 and Remark 2.2, the result holds. □

If $n=p^t,$ where t > 1, then the graph $\Gamma \prime ({\mathbb{Z}}_n)$ is a null graph. Let $n\ne p^t$ for any $t\in \mathbb{N}.$ Then by Lemma 3.5, $\Upsilon {\prime }_n$ is connected graph so that $D_{d_i}>0.$ By Theorem 4.1, out of $n-\varphi (n)-1$ Laplacian eigenvalues of $\Gamma \prime ({\mathbb{Z}}_n)$ note that $n-\varphi (n)-1-k$ eigenvalues are non-zero integers. The remaining k Laplacian eigenvalues of $\Gamma \prime ({\mathbb{Z}}_n)$ are the roots of the characteristic equation of the matrix $L(\Upsilon {\prime }_n)$ given in equation (3).

Lemma 4.2.

Let n = pq be a product of two distinct primes. Then the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_n)$ is given by $$\left(\begin{array}{cccc}0& p+q-2& p-1& q-1\\ 1& 1& q-2& p-2\end{array}\right).$$

Proof.

By Lemma 3.6, we have $\Gamma \prime ({\mathbb{Z}}_{pq})=\Upsilon {\prime }_{pq}[\Gamma \prime ({\mathcal{A}}_p),\Gamma \prime ({\mathcal{A}}_q)],$ where $\Upsilon {\prime }_{pq}=K_2,\Gamma \prime ({\mathcal{A}}_p)={\overline{K}}_{\varphi (q)}$ and $\Gamma \prime ({\mathcal{A}}_q)={\overline{K}}_{\varphi (p)}$ (cf. Lemma 3.2 and Corollary 3.4). Consequently, by Theorem 4.1, the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_{pq})$ is $$\begin{array}{cc}{\Phi }_L(\Gamma \prime ({\mathbb{Z}}_{pq}))\hfill & =(D_p+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_p))\setminus \left\{0\right\}))\hfill \\ \hfill & \cup (D_q+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_q))\setminus \left\{0\right\}))\cup \Phi (L(\Upsilon {\prime }_{pq}))\hfill \\ \hfill & =\left(\begin{array}{cc}p-1& q-1\\ q-2& p-2\end{array}\right)\cup \Phi (L(\Upsilon {\prime }_{pq})).\hfill \end{array}$$

Then the matrix $$L(\Upsilon {\prime }_{pq})=\left[\begin{array}{cc}p-1& -(p-1)\\ -(q-1)& q-1\end{array}\right]$$ has eigenvalues $p+q-2$ and 0. Thus, we have the result. □

Notations 4.3.

${({\lambda }_i)}^{[{\mu }_i]}$ denotes the eigenvalue λ_i of $\mathcal{L}(\Gamma \prime ({\mathbb{Z}}_n))$ with multiplicity μ_i.

Lemma 4.4.

For distinct primes p and q, if $n=p^{2}q$ then the Laplacian eigenvalues of $\Gamma \prime ({\mathbb{Z}}_n)$ consists of the set $\left\{{(p^2-p)}^{[(p-1)(q-1)-1]},{(pq-p)}^{[p^2-p-1]},{(p^2-1)}^{[q-2]},{(q-1)}^{[p-2]}\right\}$ and the remaining eigenvalues are the roots of the characteristic polynomial $$\begin{array}{l}{x}^4-\{(p-1)(2p+1)+(p+1)(q-1)\}{x}^3+\{p{(p-1)}^2(p+1)\\ +(p-1){(p+1)}^2(q-1)+p{(q-1)}^2+{(p-1)}^2(q-1)\}{x}^2\\ -p(p-1)(q-1)\{(p-1)(p+1)+p(q-1)\}x.\end{array}$$

Proof.

First note that $\Upsilon {\prime }_{p^{2}q}$ is the path graph given by $p\sim q\sim p^2\sim pq.$ By Lemma 3.6, $$\Gamma \prime ({\mathbb{Z}}_{p^{2}q})=\Upsilon {\prime }_{p^{2}q}[\Gamma \prime ({\mathcal{A}}_p),\Gamma \prime ({\mathcal{A}}_q),\Gamma \prime ({\mathcal{A}}_{p^2}),\Gamma \prime ({\mathcal{A}}_{pq})],$$ where $\Gamma \prime ({\mathcal{A}}_p)={\overline{K}}_{\varphi (pq)},\Gamma \prime ({\mathcal{A}}_q)={\overline{K}}_{\varphi (p^2)},\Gamma \prime ({\mathcal{A}}_{p^2})={\overline{K}}_{\varphi (q)}$ and $\Gamma \prime ({\mathcal{A}}_{pq})={\overline{K}}_{\varphi (p)}.$ It follows that $D_p=\varphi (p^2)=p^2-p$ and $D_q=\varphi (pq)+\varphi (q)=p(q-1),D_{p^2}=\varphi (p^2)+\varphi (p)=p^2-1$ and $D_{pq}=\varphi (q)=q-1.$ Therefore, by Theorem 4.1, the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_{p^{2}q})$ is $$\begin{array}{ll}{\Phi }_L(\Gamma \prime ({\mathbb{Z}}_{p^{2}q}))\hfill & =(D_p+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_p))\setminus \left\{0\right\}))\cup (D_q+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_q))\setminus \left\{0\right\}))\cup (D_{p^2}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^2}))\setminus \left\{0\right\}))\hfill \\ \hfill & \cup (D_{pq}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{pq}))\setminus \left\{0\right\}))\cup \Phi (L(\Upsilon {\prime }_{p^{2}q}))\hfill \\ \hfill & =\left(\begin{array}{cccc}{p}^2-p& pq-p& p^2-1& q-1\\ (p-1)(q-1)-1& p^2-p-1& q-2& p-2\end{array}\right)\cup \Phi (L(\Upsilon {\prime }_{p^{2}q})).\hfill \end{array}$$

Thus, the remaining Laplacian eigenvalues can be obtained by the characteristic polynomial (given in the statement) of the matrix $$L(\Upsilon {\prime }_{p^{2}q})=\left[\begin{array}{cccc}{p}^2-p& -p^2+p& 0& 0\\ -(p-1)(q-1)& p(q-1)& -(q-1)& 0\\ 0& -p^2+p& p^2-1& -p+1\\ 0& 0& -q+1& q-1\end{array}\right].$$

□

Lemma 4.5.

For distinct primes p and q, if $n=p^{n_1}q$ then the Laplacian eigenvalues of $\Gamma \prime ({\mathbb{Z}}_n)$ consists of the set $\{{(\varphi (p^{n_1}))}^{[\varphi (p^{n_1-1}q)-1]},{(\varphi (p^{n_1})+\varphi (p^{n_1-1}))}^{[\varphi (p^{n_1-2}q)-1]},{({\sum }_{i=0}^2\varphi (p^{n_1-i}))}^{[\varphi (p^{n_1-3}q)-1]},\dots ,{({\sum }_{i=0}^{n_1-1}\varphi (p^{n_1-i}))}^{[\varphi (p^{n_1-n_1}q)-1]},{({\sum }_{i=1}^{n_1}\varphi (p^{n_1-i}q))}^{[\varphi (p^{n_1})-1]},{({\sum }_{i=2}^{n_1}\varphi (p^{n_1-i}q))}^{[\varphi (p^{n_1-1})-1]},\dots ,{(\varphi (q))}^{[\varphi (p)-1]}\}$ and the remaining eigenvalues are the eigenvalues of the matrix given in equation (3).

Proof.

Note that $\{p,p^2,\dots ,p^{n_1},q,pq,p^{2}q,\dots ,p^{n_1-1}q\}$ is the vertex set of the graph $\Upsilon {\prime }_{p^{n_1}q}.$ By Lemma 3.6, $$\Gamma \prime ({\mathbb{Z}}_{p^{n_1}q})=\Upsilon {\prime }_{p^{n_1}q}[\Gamma \prime ({\mathcal{A}}_p),\Gamma \prime ({\mathcal{A}}_{p^2}),\dots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1}}),\Gamma \prime ({\mathcal{A}}_q),\Gamma \prime ({\mathcal{A}}_{pq}),\Gamma \prime ({\mathcal{A}}_{p^{2}q}),\dots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1-1}q})],$$ where, $\Gamma \prime ({\mathcal{A}}_p)={\overline{K}}_{\varphi (p^{n_1-1}q)},\Gamma \prime ({\mathcal{A}}_{p^2})={\overline{K}}_{\varphi (p^{n_1-2}q)},\cdots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1}})={\overline{K}}_{\varphi (q)},\Gamma \prime ({\mathcal{A}}_q)={\overline{K}}_{\varphi (p^{n_1})},\Gamma \prime ({\mathcal{A}}_{pq})={\overline{K}}_{\varphi (p^{n_1-1})},\cdots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1-1}q})={\overline{K}}_{\varphi (p)}.$ It follows that $D_p=\varphi (p^{n_1}),D_{p^2}=\varphi (p^{n_1})+\varphi (p^{n_1-1}),\cdots ,D_{p^{n_1}}={\sum }_{i=0}^{n_1-1}\varphi (p^{n_1-i}),D_q={\sum }_{i=1}^{n_1}\varphi (p^{n_1-i}q),D_{pq}={\sum }_{i=2}^{n_1}\varphi (p^{n_1-i}q),\cdots ,D_{p^{n_1-1}q}=\varphi (q).$ Consequently, by Theorem 4.1, the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_{p^{n_1}q})$ is $$\begin{array}{cc}{\Phi }_L(\Gamma \prime ({\mathbb{Z}}_{p^{n_1}q}))\hfill & =(D_p+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_p))\setminus \left\{0\right\}))\cup (D_{p^2}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^2}))\setminus \left\{0\right\}))\cup \cdots \cup (D_{p^{n_1}}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^{n_1}}))\setminus \left\{0\right\}))\hfill \\ \hfill & \cup (D_q+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_q))\setminus \left\{0\right\}))\cup (D_{pq}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{pq}))\setminus \left\{0\right\}))\cup \cdots \cup (D_{p^{n_1-1}q}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^{n_1-1}q}))\setminus \left\{0\right\}))\hfill \\ \hfill & \cup \Phi (L(\Upsilon {\prime }_{p^{n_1}q})).\hfill \\ \hfill & =\left(\begin{array}{ccccccc}\varphi (p^{n_1})& \cdots & \sum _{i=0}^{n_1-1}\varphi (p^{n_1-i})& \sum _{i=1}^{n_1}\varphi (p^{n_1-i}q)& \sum _{i=2}^{n_1}\varphi (p^{n_1-i}q)& \cdots & \varphi (q)\\ \varphi (p^{n_1-1}q)-1& \cdots & \varphi (p^{n_1-n_1}q)-1& \varphi (p^{n_1})-1& \varphi (p^{n_1})-1& \cdots & \varphi (p)-1\end{array}\right)\cup \Phi (L(\Upsilon {\prime }_{p^{n_1}q})).\hfill \end{array}$$

Thus, the remaining $2n_1$ Laplacian eigenvalues are the eigenvalues of the matrix $L(\Upsilon {\prime }_{p^{n_1}q})=$ $$\left[\begin{array}{ccccccccc}\varphi (p^{n_1})& 0& 0& \cdots & 0& \cdots & \cdots & 0& -\varphi (p^{n_1})\\ 0& \sum _{i=0}^1\varphi (p^{n_1-i})& 0& \cdots & \cdots & 0& \cdots & -\varphi (p^{n_1-1})& -\varphi (p^{n_1})\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \cdots & \sum _{i=0}^{n_1-1}\varphi (p^{n_1-i})& -\varphi (p)& -\varphi (p^2)& \cdots & -\varphi (p^{n_1})\\ 0& 0& 0& \cdots & -\varphi (q)& \varphi (q)& 0& \cdots & 0\\ 0& \cdots & \cdots & -\varphi (pq)& -\varphi (q)& 0& \varphi (pq)+\varphi (q)& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -\varphi (p^{n_1-1}q)& -\varphi (p^{n_1-2}q)& \cdots & \cdots & -\varphi (q)& 0& 0& \cdots & \sum _{i=1}^{n_1}\varphi (p^{n_1-i}q)\end{array}\right]$$ where matrix $L(\Upsilon {\prime }_{p^{n_1}q})$ is obtained by indexing the rows and columns as $p,p^2,\dots ,p^{n_1},p^{n_1-1}q,\dots ,pq,q.$ □

Theorem 4.6.

If $n=p^{n_1}{q}^{n_2}$, where p and q are distinct primes. Then the set of Laplacian eigenvalues of $\Gamma \prime ({\mathbb{Z}}_n)$ consists of $$\begin{array}{c}\begin{array}{c}{\left(\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})\right)}^{[\varphi (p^{n_1-1}{q}^{n_2})-1]}, {\left(\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-1}{q}^{n_2-i})\right)}^{[\varphi (p^{n_1-2}{q}^{n_2})-1]},\hfill \\ {\left(\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-2}{q}^{n_2-i})\right)}^{[\varphi (p^{n_1-3}{q}^{n_2})-1]},\hfill \\ ⋮\hfill \\ {\left(\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-1}{q}^{n_2-i})+\cdots +\sum _{i=1}^{n_2}\varphi (p{q}^{n_2-i})\right)}^{[\varphi (q^{n_2})-1]},\hfill \end{array}\\ \begin{array}{c}{\left(\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})\right)}^{[\varphi (p^{n_1}{q}^{n_2-1})-1]},{\left(\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-1})\right)}^{[\varphi (p^{n_1}{q}^{n_2-2})-1]},\hfill \\ {\left(\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-1})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-2})\right)}^{[\varphi (p^{n_1}{q}^{n_2-3})-1]},\hfill \\ ⋮\hfill \\ {\left(\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-1})+\cdots +\sum _{i=1}^{n_1}\varphi (p^{n_1-i}q)\right)}^{[\varphi (p^{n_1})-1]},\hfill \\ {\left(\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=2}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})\right)}^{[\varphi (p^{n_1-1}{q}^{n_2-1})-1]},\hfill \end{array}\\ \begin{array}{c}{\left(\sum _{i=3}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=2}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-1}{q}^{n_2-i})\right)}^{[\varphi (p^{n_1-2}{q}^{n_2-1})-1]},\hfill \\ ⋮\hfill \\ {\left(\sum _{i=2}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-2})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-3})+\cdots +\sum _{i=1}^{n_1-1}\varphi (p^{n_1-i})\right)}^{[\varphi (q^{n_2-1})-1]},\hfill \\ {\left(\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=3}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-1})\right)}^{[\varphi (p^{n_1-1}{q}^{n_2-2})-1]},\hfill \\ ⋮\hfill \\ {\left(\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=1}^{n_2}\varphi (p^{n_1-2}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-3}{q}^{n_2-i})+\cdots +\sum _{i=1}^{n_2-1}\varphi (q^{n_2-i})\right)}^{[\varphi (p^{n_1-1})-1]},\hfill \\ ⋮\hfill \\ {\left(\sum _{i=1}^{n_2-1}\varphi (q^{n_2-i})+\varphi (q^{n_2})\right)}^{[\varphi (p)-1]}\hfill \end{array}\end{array}$$ and the remaining $(n_1+1)(n_2+1)-2$ eigenvalues are given by the zeros of the characteristic polynomial of the matrix given in equation (3).

Proof.

The set of proper divisors of $n=p^{n_1}{q}^{n_2}$ is $$\{p,p^2,\dots ,p^{n_1},q,q^2,\dots ,q^{n_2},pq,p^{2}q,\dots ,p^{n_1}q,p{q}^2,p^2{q}^2,\dots ,p^{n_1}{q}^2,\cdots ,p{q}^{n_2},p^2{q}^{n_2},\dots ,p^{n_1-1}{q}^{n_2}\}.$$

By the definition of $\Upsilon {\prime }_n,$ note that

• $p^i\sim q^j$ for all i, j.

• $p^i\sim p^{i_1}{q}^{j_1}$ for $i>i_1$ and $j_1>0.$

• $q^j\sim p^i{q}^{j_1}$ for $j>j_1$ and i > 0.

• If either $i_1>i_2,j_1<j_2$ or $j_1>j_2,i_1<i_2,$ then $p^{i_1}{q}^{j_1}\sim p^{i_2}{q}^{j_2}.$

In view of Lemma 3.6, $$\begin{array}{c}\Gamma \prime ({\mathbb{Z}}_{p^{n_1}{q}^{n_2}})=\Upsilon {\prime }_{p^{n_1}{q}^{n_2}}[\Gamma \prime ({\mathcal{A}}_p),\Gamma \prime ({\mathcal{A}}_{p^2}),\dots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1}}),\Gamma \prime ({\mathcal{A}}_q),\Gamma \prime ({\mathcal{A}}_{q^2}),\dots ,\Gamma \prime ({\mathcal{A}}_{q^{n_2}}),\Gamma \prime ({\mathcal{A}}_{pq}),\Gamma \prime ({\mathcal{A}}_{p^{2}q}),\dots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1}q}),\\ \cdots ,\Gamma \prime ({\mathcal{A}}_{p{q}^{n_2}}),\dots ,\Gamma \prime ({\mathcal{A}}_{p^{n_1-1}{q}^{n_2}})].\end{array}$$

Therefore, by Lemma 3.2 and Corollary 3.4, we get

$\Gamma \prime ({\mathcal{A}}_{p^i})={\overline{K}}_{\varphi (p^{n_1-i}{q}^{n_2})},$ where $1\le i\le n_1,$

$\Gamma \prime ({\mathcal{A}}_{q^j})={\overline{K}}_{\varphi (p^{n_1}{q}^{n_2-j})}$ where $1\le j\le n_2,$

$\Gamma \prime ({\mathcal{A}}_{p^i{q}^j})={\overline{K}}_{\varphi (p^{n_1-i}{q}^{n_2-j})}.$

Consequently, we have $$\begin{array}{c}\begin{array}{c}{D}_p=\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i}),D_{p^2}=\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-1}{q}^{n_2-i}),\hfill \\ ⋮\hfill \\ D_{p^{n_1}}=\sum _{i=1}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-1}{q}^{n_2-i})+\cdots +\sum _{i=1}^{n_2}\varphi (p{q}^{n_2-i}),\hfill \\ D_q=\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2}),\hfill \\ ⋮\hfill \\ D_{q^{n_2}}=\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-1})+\cdots +\sum _{i=1}^{n_1}\varphi (p^{n_1-i}q),D_{pq}=\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=2}^{n_2}\varphi (p^{n_1}{q}^{n_2-i}),\hfill \\ ⋮\hfill \end{array}\\ \begin{array}{l}{D}_{p^{n_1}q}=\sum _{i=2}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-2})+\sum _{i=1}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-3})+\cdots +\sum _{i=1}^{n_1-1}\varphi (p^{n_1-i}),\hfill \\ D_{p{q}^2}=\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=3}^{n_2}\varphi (p^{n_1}{q}^{n_2-i})+\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2-1}),\hfill \\ ⋮\hfill \\ D_{p{q}^{n_2}}=\sum _{i=2}^{n_1}\varphi (p^{n_1-i}{q}^{n_2})+\sum _{i=1}^{n_2}\varphi (p^{n_1-2}{q}^{n_2-i})+\sum _{i=1}^{n_2}\varphi (p^{n_1-3}{q}^{n_2-i})+\cdots +\sum _{i=1}^{n_2-1}\varphi (q^{n_2-i}),\hfill \\ ⋮\hfill \\ D_{p^{n_1-1}{q}^{n_2}}=\sum _{i=1}^{n_2-1}\varphi (q^{n_2-i})+\varphi (q^{n_2}).\hfill \end{array}\end{array}$$

Therefore, by Theorem 4.1, the Laplacian spectrum of $\Gamma \prime ({\mathbb{Z}}_{p^{n_1}{q}^{n_2}})$ is $$\begin{array}{cc}{\Phi }_L(\Gamma \prime ({\mathbb{Z}}_{p^{n_1}{q}^{n_2}}))\hfill & =(D_p+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_p))\setminus \left\{0\right\}))\cup (D_{p^2}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^2}))\setminus \left\{0\right\}))\cup \cdots \cup (D_{p^{n_1}}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^{n_1}}))\setminus \left\{0\right\}))\hfill \\ \hfill & \cup (D_q+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_q))\setminus \left\{0\right\}))\cup (D_{q^2}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{q^2}))\setminus \left\{0\right\}))\cup \cdots \cup (D_{q^{n_2}}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{q^{n_2}}))\setminus \left\{0\right\}))\hfill \\ \hfill & \cup (D_{pq}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{pq}))\setminus \left\{0\right\}))\cup \cdots \cup (D_{p^{n_1}q}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^{n_1}q}))\setminus \left\{0\right\}))\cup \cdots \hfill \\ \hfill & \cup (D_{p{q}^{n_2}}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p{q}^{n_2}}))\setminus \left\{0\right\}))\cup \cdots \cup (D_{p^{n_1-1}{q}^{n_2}}+({\Phi }_L(\Gamma \prime ({\mathcal{A}}_{p^{n_1-1}{q}^{n_2}}))\setminus \left\{0\right\}))\cup \Phi (L(\Upsilon {\prime }_{p^{n_1}{q}^{n_2}})).\hfill \end{array}$$

The remaining $(n_1+1)(n_2+1)-2$ eigenvalues are the zeros of the characteristic polynomial of the matrix $L(\Upsilon {\prime }_{p^{n_1}{q}^{n_2}})$ given in equation (3). □

5. The Laplacian spectral radius and the algebraic connectivity of $\Gamma \prime ({\mathbb{Z}}_n)$

In this section, we study the algebraic connectivity and the Laplacian spectral radius of $\Gamma \prime ({\mathbb{Z}}_n).$ We obtain all those values of n for which the Laplacian spectral radius of $\Gamma \prime ({\mathbb{Z}}_n)$ is equal to order of $\Gamma \prime ({\mathbb{Z}}_n).$ Moreover, the values of n for which the algebraic connectivity and the vertex connectivity coincide are also described. The following theorem follows from the relation $\lambda (\Gamma )=|V(\Gamma )|-\mu (\overline{\Gamma })$ and the fact $\overline{\Gamma }$ is disconnected if and only if Γ is the join of two graphs.

Theorem 5.1

([13]). If Γ is a graph on m vertices, then $\lambda (\Gamma )\le m$. Further, equality holds if and only if $\overline{\Gamma }$ is disconnected if and only if Γ is the join of two graphs.

In view of Theorem 5.1, first we characterize the values of n for which the complement of $\Gamma \prime ({\mathbb{Z}}_n)$ is disconnected.

Proposition 5.2.

$\overline{\Gamma \prime ({\mathbb{Z}}_n)}$ is disconnected if and only if n is a product of two distinct primes.

Proof.

Let p and q be two distinct primes. If n = pq, then by Remark 3.1 we get $V(\Gamma \prime ({\mathbb{Z}}_n))={\mathcal{A}}_p\cup {\mathcal{A}}_q$ such that ${\mathcal{A}}_p\cap {\mathcal{A}}_q=\varnothing .$ In fact, $\Gamma \prime ({\mathbb{Z}}_n)=K_{\varphi (q),\varphi (p)}$ is a complete bipartite graph. Consequently, $\overline{\Gamma \prime ({\mathbb{Z}}_n)}$ is a disconnected graph.

Conversely, suppose $\overline{\Gamma \prime ({\mathbb{Z}}_n)}$ is disconnected. Clearly, for n = p  there is nothing to prove. If $n=p^{\alpha }$ for some $1<\alpha \in \mathbb{N},$ then $\Gamma \prime ({\mathbb{Z}}_{p^{\alpha }})$ is a null graph. Consequently, $\overline{\Gamma \prime ({\mathbb{Z}}_{p^{\alpha }})}$ is a complete graph which is not possible. If possible, let $n\ne pq.$ Let d₁ and d₂ be the proper divisors of n and let $x\in {\mathcal{A}}_{d_1},y\in {\mathcal{A}}_{d_2}.$ If d₁ = d₂ then clearly $x\sim y$ in $\overline{\Gamma \prime ({\mathbb{Z}}_n)}.$ If $d_1\ne d_2$ such that either $d_1|d_2$ or $d_2|d_1$ then $x\sim y$ in $\overline{\Gamma \prime ({\mathbb{Z}}_n)}$ (cf. Lemma 3.3). If $d_1\ne d_2$ and neither $d_1|d_2$ nor $d_2|d_1,$ then there exist two primes p₁ and p₂ such that $p_1|d_1$ and $p_2|d_2.$ Consequently, $x\sim z_1\sim z_2\sim z_3\sim y$ in $\overline{\Gamma \prime ({\mathbb{Z}}_n)}$ for some $z_1\in {\mathcal{A}}_{p_1},z_2\in {\mathcal{A}}_{p_1{p}_2}$ and $z_3\in {\mathcal{A}}_{p_2}.$ Thus, $\overline{\Gamma \prime ({\mathbb{Z}}_n)}$ is connected; a contradiction. Hence, n must be a product of two distinct primes. □

Since $|V(\Gamma \prime ({\mathbb{Z}}_n))|=n-\varphi (n)-1,$ by using the Proposition 5.2 in Theorem 5.1, we have the following proposition.

Proposition 5.3.

$\lambda (\Gamma \prime ({\mathbb{Z}}_n))=|V(\Gamma \prime ({\mathbb{Z}}_n))|$ if and only if n is a product of two distinct primes. Moreover, if n = pq then $\lambda (\Gamma \prime ({\mathbb{Z}}_n))=p+q-2.$

Now we classify all those values of n for which the algebraic connectivity and the vertex connectivity of $\Gamma \prime ({\mathbb{Z}}_n)$ are equal. The following theorem is useful in this study.

Theorem 5.4.

[15] Let Γ be a non-complete connected graph on m vertices. Then $\kappa (\Gamma )=\mu (\Gamma )$ if and only if Γ can be written as ${\Gamma }_1\vee {\Gamma }_2$, where ${\Gamma }_1$ is a disconnected graph on $m-\kappa (\Gamma )$ vertices and ${\Gamma }_2$ is a graph on $\kappa (\Gamma )$ vertices with $\mu ({\Gamma }_2)\ge 2\kappa (\Gamma )-m.$

Lemma 5.5.

For distinct primes p and q, if n = pq where p < q then $\kappa (\Gamma \prime ({\mathbb{Z}}_n))=\delta (\Gamma \prime ({\mathbb{Z}}_n))=p-1.$

Proof.

For n = pq, $\Gamma \prime ({\mathbb{Z}}_n)$ is a complete bipartite graph with partition sets ${\mathcal{A}}_p$ and ${\mathcal{A}}_q.$ Hence, $\kappa (\Gamma \prime ({\mathbb{Z}}_n))=\delta (\Gamma \prime ({\mathbb{Z}}_n))=\text{min}\left\{\right|{\mathcal{A}}_p|,|{\mathcal{A}}_q\left|\right\}=p-1$ □

Theorem 5.6.

For the graph $\Gamma \prime ({\mathbb{Z}}_n)$, we have $\mu (\Gamma \prime ({\mathbb{Z}}_n))\le \kappa (\Gamma \prime ({\mathbb{Z}}_n))$. The equality holds if and only if n is a product of two distinct primes.

Proof.

By [15], for any graph Γ which is not complete, we have $\mu (\Gamma )\le \kappa (\Gamma ).$ If n = 4 then there is nothing to prove because $\Gamma \prime ({\mathbb{Z}}_4)$ is the graph of one vertex only. If $n\ne 4$ then $\Gamma \prime ({\mathbb{Z}}_n)$ is not a complete graph. Consequently, $\mu (\Gamma \prime ({\mathbb{Z}}_n))\le \kappa (\Gamma \prime ({\mathbb{Z}}_n)).$

If n is not a product of two distinct primes then by Proposition 5.2 and by Theorem 5.1, $\Gamma \prime ({\mathbb{Z}}_n)$ cannot be written as the join of two graphs. Thus, by Theorem 5.4, we obtain $\mu (\Gamma \prime ({\mathbb{Z}}_n))<\kappa (\Gamma \prime ({\mathbb{Z}}_n)).$ If n = pq, where p and q are distinct primes such that p <q  , then by Theorem 5.1, Proposition 5.2, Theorem 5.4 and Lemma 5.5, we obtain $\mu (\Gamma \prime ({\mathbb{Z}}_n))=\kappa (\Gamma \prime ({\mathbb{Z}}_n))=p-1.$ □

6. The Wiener index of $\Gamma \prime ({\mathbb{Z}}_n)$

In this section, we obtain the Wiener index of the cozero-divisor graph of the ring ${\mathbb{Z}}_n$ for arbitrary $n\in \mathbb{N}.$ Consequently, we obtain the diameter of $\Gamma \prime ({\mathbb{Z}}_n)$ (see Proposition 6.4). For a prime p and $1\le \alpha \in \mathbb{N},$ the graph $\Gamma \prime ({\mathbb{Z}}_p)$ is empty whereas $\Gamma \prime ({\mathbb{Z}}_{p^{\alpha }})$ is a null graph. Therefore $W(\Gamma \prime ({\mathbb{Z}}_p))=W(\Gamma \prime ({\mathbb{Z}}_{p^{\alpha }}))=0.$

Theorem 6.1.

For $1\le i\le \tau (n)-2$, let d_i’s be the proper divisors of n. If $n=p_1{p}_2\cdots p_k$, where p_i’s are distinct primes and $2\le k\in \mathbb{N}$, then $$W(\Gamma \prime ({\mathbb{Z}}_n))=\sum _{i=1}^{2^k-2}\varphi \left(\frac{n}{d_i}\right)\left(\varphi \left(\frac{n}{d_i}\right)-1\right)+\frac{1}{2}\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\varphi \left(\frac{n}{d_i}\right)\varphi \left(\frac{n}{d_j}\right)+2\sum _{\begin{array}{c}{d}_i|d_j\\ i\ne j\end{array}}\varphi \left(\frac{n}{d_i}\right)\varphi \left(\frac{n}{d_j}\right).$$

Proof.

To determine the Wiener index of $\Gamma \prime ({\mathbb{Z}}_n),$ we first obtain the distances between the vertices of each ${\mathcal{A}}_{d_i}$ and two distinct ${\mathcal{A}}_{d_i}$’s, respectively. For a proper divisor d_i of n, let $x,y\in {\mathcal{A}}_{d_i}.$ Since $\Gamma \prime ({\mathbb{Z}}_n)$ is connected, by Corollary 3.4, there exists a proper divisor d_r of n such that $x\sim z$ for each $x\in {\mathcal{A}}_{d_i}$ and $z\in {\mathcal{A}}_{d_r}.$ Consequently, $d(x,y)=2$ for any two distinct $x,y\in {\mathcal{A}}_{d_i}.$ Now we obtain the distances between the vertices of any two distinct ${\mathcal{A}}_{d_i}$’s through the following cases.

Case-1: Neither $d_i|d_j$ nor $d_j|d_i.$ By Lemma 3.3, $d(x,y)=1$ for every $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$

Case-2: $d_i|d_j.$ For $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}$ we have $x\nsim y.$ Without loss of generality, assume that $d_j=p_1{p}_2\cdots p_m{d}_i,$ where $1\le m\le k-2.$ Since d_j is a proper divisor of n there exists a prime p such that $p\nmid d_j.$ Consequently, $p\nmid d_i.$ It follows that for $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}$ there exists a $z\in {\mathcal{A}}_p$ such that $x\sim z\sim y.$ Thus, $d(x,y)=2$ for each $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$

Thus, in view of all the possible distances between the vertices of $\Gamma \prime ({\mathbb{Z}}_n),$ we get $$\begin{array}{ll}W(\Gamma \prime ({\mathbb{Z}}_n))\hfill & =\frac{1}{2}\sum _{u\in V(\Gamma \prime ({\mathbb{Z}}_n))}\sum _{v\in V(\Gamma \prime ({\mathbb{Z}}_n))}d(u,v)\hfill \\ \hfill & =\frac{1}{2}[\sum _{i=1}^{2^k-2}2|{\mathcal{A}}_{d_i}|(\left|{\mathcal{A}}_{d_i}\right|-1)+\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|]\hfill \\ \hfill & +2\sum _{}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill \\ \hfill & =\sum _{i=1}^{2^k-2}\varphi (\frac{n}{d_i})(\varphi (\frac{n}{d_i})-1)\hfill \\ \hfill & +\frac{1}{2}\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\varphi (\frac{n}{d_i})\varphi (\frac{n}{d_j})\hfill \\ \hfill & +2\sum _{\begin{array}{c}{d}_i|d_j\\ i\ne j\end{array}}\varphi (\frac{n}{d_i})\varphi (\frac{n}{d_j}).\hfill \end{array}$$ □

Corollary 6.2.

If n = pq, where p, q are distinct primes, then $W(\Gamma \prime ({\mathbb{Z}}_n))=(p-1)(q-1)+(p-1)(p-2)+(q-1)(q-2).$

Theorem 6.3.

Let $n=p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}\cdots p_k^{n_k}$ with $k\ge 2$, where p_i’s are distinct primes and let $D=\{d_1,d_2,\dots ,d_{\tau (n)-2}\}$ be the set of all proper divisors of n. For $d_i|d_j$, define $$\begin{array}{cc}A\hfill & =\{(d_i,d_j)\in D\times D|d_i\ne p_r^s\};\hfill \\ B\hfill & =\{(d_i,d_j)\in D\times D|d_i=p_r^s\mathit{and}\frac{n}{d_j}\ne p_r^t\};\hfill \\ C\hfill & =\{(d_i,d_j)\in D\times D|d_i=p_r^s\mathit{and}\frac{n}{d_j}=p_r^t\}.\hfill \end{array}$$ Then $$\begin{array}{c}W(\Gamma \prime ({\mathbb{Z}}_n))=\sum _{i=1}^{\tau (n)-2}\varphi \left(\frac{n}{d_i}\right)\left(\varphi \left(\frac{n}{d_i}\right)-1\right)+\frac{1}{2}\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\varphi \left(\frac{n}{d_i}\right)\varphi \left(\frac{n}{d_j}\right)+2\sum _{(d_i,d_j)\in A}\varphi \left(\frac{n}{d_i}\right)\varphi \left(\frac{n}{d_j}\right)+2\sum _{(d_i,d_j)\in B}\varphi \left(\frac{n}{d_i}\right)\varphi \left(\frac{n}{d_j}\right)\hfill \\ + 3\sum _{(d_i,d_j)\in C}\varphi \left(\frac{n}{d_i}\right)\varphi \left(\frac{n}{d_j}\right).\hfill \end{array}$$

Proof.

In view of Remark 3.1, first we obtain all the possible distances between the vertices of ${\mathcal{A}}_{d_i}$ and ${\mathcal{A}}_{d_j},$ where d_i and d_j are proper divisors of n. If i = j  then by the proof of Theorem 6.1, we get $d(x,y)=2$ for any two distinct $x,y\in {\mathcal{A}}_{d_i}.$ Now suppose that $i\ne j.$ If $d_i\nmid d_j$ and $d_j\nmid d_i,$ then by Lemma 3.3, we get $d(x,y)=1$ for every $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$ If $d_i|d_j$ then we obtain the possible distances through the following cases.

Case-1: $(d_i,d_j)\in A.$ Since $d_i|d_j,$ we have $x\nsim y$ for any $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$ Note that $d_i\ne p_r^s$ implies that $d_i=p_1^{{\beta }_1}{p}_2^{{\beta }_2}\cdots p_m^{{\beta }_m}$ for some β_i’s $\in \mathbb{N}\cup \left\{0\right\}$ and $m\ge 2.$ Consequently, $d_j=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_m^{{\alpha }_m}$ for some α_i’s $\in \mathbb{N}\cup \left\{0\right\}.$ Since d_j is a proper divisor of n there exists $l\in \{1,2,\dots ,k\}$ such that $p_l^{n_l}\nmid d_j.$ Also, $p_l^{n_l}\nmid d_i.$ Further, $m\ge 2$ follows that $d_i\nmid p_l^{n_l}$ and $d_j\nmid p_l^{n_l}.$ Now for any $x\in {\mathcal{A}}_{d_i},y\in {\mathcal{A}}_{d_j}$ there exists a $z\in {\mathcal{A}}_{p_l^{n_l}}$ such that $x\sim z\sim y.$ Thus $d(x,y)=2$ for every $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$

Case-2: $d_i=p_r^s$ for some $r\in \{1,2,\dots ,k\}$ and $1\le s\le n_r.$ Suppose $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$ Then we obtain d(x, y) in the following subcases:

Subcase-2.1: $(d_i,d_j)\in B.$ Suppose $d_j=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r}\cdots p_k^{{\alpha }_k}.$ Since $\frac{n}{d_j}\ne p_r^t$ there exists a prime $p_m\in \{p_1,p_2,\dots ,p_k\}\setminus \left\{p_r\right\}$ and ${\alpha }_m\le n_m$ such that $p_m^{n_m}\nmid d_j.$ Consequently, $p_m^{n_m}\nmid d_i.$ Moreover, $d_i\nmid p_m^{n_m}$ and $d_j\nmid p_m^{n_m}.$ Thus, for every $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j},$ we get $x\sim z\sim y$ for some $z\in {\mathcal{A}}_{p_m^{n_m}}.$ Hence, $d(x,y)=2$ for each $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$

Subcase-2.2: $(d_i,d_j)\in C.$ Then $d_j=p_1^{n_1}{p}_2^{n_2}\cdots p_r^{{\alpha }_r}\cdots p_k^{n_k},$ where $n_r-{\alpha }_r=t\ge 1.$ Since $d_i|d_j,$ for each $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j},$ we have $d(x,y)\ge 2$ (cf. Lemma 3.3). First, we show that $d(x,y)>2$ for any $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$ In this connection, it is sufficient to prove that for any proper divisor d of n, we have either $d|d_j$ or $d_i|d.$ Suppose that $d\nmid d_j.$ Then $d=p_1^{{\gamma }_1}{p}_2^{{\gamma }_2}\cdots p_r^{{\gamma }_r}\cdots p_k^{{\gamma }_k}$ together with ${\gamma }_r>{\alpha }_r.$ Since $p_r^s=d_i|d_j,$ we get $s\le {\alpha }_r<{\gamma }_r.$ Consequently, $d_i|d.$

Since $n=p_1^{n_1}{p}_2^{n_2}\cdots p_r^{n_r}\cdots p_k^{n_k}$ with $k\ge 2,$ there exists a prime $q\ne p_r$ such that $q|n.$ Clearly, $q\nmid d_i$ and $d_i\nmid q.$ Also, $p_r^{n_r}\nmid q$ and $q\nmid p_r^{n_r}.$ Since ${\alpha }_r<n_r,$ we obtain $d_j\nmid p_r^{n_r}$ and $p_r^{n_r}\nmid d_j.$ Thus, in view of Lemma 3.3, for any $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j},$ there exist $z\in {\mathcal{A}}_q$ and $w\in {\mathcal{A}}_{p_r^{n_r}}$ such that $x\sim z\sim w\sim y.$ Hence, $d(x,y)=3$ for every $x\in {\mathcal{A}}_{d_i}$ and $y\in {\mathcal{A}}_{d_j}.$ In view of the cases and arguments discussed in this proof, we have $$\begin{array}{l}W(\Gamma \prime ({\mathbb{Z}}_n))=\frac{1}{2}\sum _{u\in V(\Gamma \prime ({\mathbb{Z}}_n))}{\sum }_{v\in V(\Gamma \prime ({\mathbb{Z}}_n)}d(u,v)\\ \begin{array}{cc}\hfill & =\frac{1}{2}[\sum _{i=1}^{\tau (n)-2}2|{\mathcal{A}}_{d_i}|(\left|{\mathcal{A}}_{d_i}\right|-1)+\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|]+2\sum _{(d_i,d_j)\in A}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|+2\sum _{(d_i,d_j)\in B}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill \\ \hfill & +3\sum _{(d_i,d_j)\in C}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill \\ \hfill & =\sum _{i=1}^{\tau (n)-2}\varphi (\frac{n}{d_i})(\varphi (\frac{n}{d_i})-1)+\frac{1}{2}\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\varphi (\frac{n}{d_i})\varphi (\frac{n}{d_j})+2\sum _{(d_i,d_j)\in A}\varphi (\frac{n}{d_i})\varphi (\frac{n}{d_j})\hfill \\ \hfill & +2\sum _{(d_i,d_j)\in B}\varphi (\frac{n}{d_i})\varphi (\frac{n}{d_j})+3\sum _{(d_i,d_j)\in C}\varphi (\frac{n}{d_i})\varphi (\frac{n}{d_j}).\hfill \end{array}\end{array}$$ □

Based on all the possible distances obtained in this section, the following proposition is easy to observe.

Proposition 6.4.

The diameter of $\Gamma \prime ({\mathbb{Z}}_n)$ is given below: $$\text{diam}(\Gamma \prime ({\mathbb{Z}}_n))=\{\begin{array}{cc}0\hfill & n=4,\hfill \\ 2\hfill & n=p_1{p}_2\cdots p_k, k\ge 2\hfill \\ 3\hfill & \text{otherwise}.\hfill \end{array}$$

Now we conclude this paper with an illustration of Theorem 6.3 for n = 72.

Example 6.5.

Consider $n=2^3·3^2=72.$ Then the number of proper divisor $\tau (n)$ of n is ${\prod }_{i=1}^k(n_i+1)-2=10.$ Therefore, $D=\{2,2^2,2^3,3,3^2,2·3,2^2·3,2^3·3,2·3^2,2^2·3^2\}.$ Let $d_1=2,d_2=2^2,d_3=2^3,d_4=3,d_5=3^2,d_6=2·3,d_7=2^2·3,d_8=2^3·3,d_9=2·3^2,d_{10}=2^2·3^2.$ By Lemma 3.2, we obtain $\left|{\mathcal{A}}_{d_1}\right|=12,\left|{\mathcal{A}}_{d_2}\right|=6,\left|{\mathcal{A}}_{d_3}\right|=6,\left|{\mathcal{A}}_{d_4}\right|=8,\left|{\mathcal{A}}_{d_5}\right|=4,\left|{\mathcal{A}}_{d_6}\right|=4,\left|{\mathcal{A}}_{d_7}\right|=2,\left|{\mathcal{A}}_{d_8}\right|=2,\left|{\mathcal{A}}_{d_9}\right|=2,\left|{\mathcal{A}}_{d_{10}}\right|=1.$ Now $$\frac{1}{2}\sum _{i=1}^{10}2\left|{\mathcal{A}}_{d_i}\right|\left(\left|{\mathcal{A}}_{d_i}\right|-1\right)=[132+30+30+56+12+12+2+2+2+0]=278$$ and $$\frac{1}{2}\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|=[96+48+48+24+24+12+48+24+24+12+12+6+16+8+8+4+4+2]=420$$ The sets A, B and C defined in Theorem 6.3 are $$\begin{array}{cc}A\hfill & =\{(d_6,d_7),(d_6,d_8),(d_6,d_9),(d_6,d_{10}),(d_7,d_8),(d_7,d_{10}),(d_9,d_{10})\};\hfill \\ B\hfill & =\{(d_1,d_2),(d_1,d_3),(d_1,d_6),(d_1,d_7),(d_1,d_8),(d_2,d_3),(d_2,d_7),(d_2,d_8),(d_3,d_8),(d_4,d_5),(d_4,d_6),(d_4,d_7),\hfill \\ \hfill & (d_4,d_9),(d_4,d_{10}),(d_5,d_9),(d_5,d_{10})\};\hfill \\ C\hfill & =\{(d_1,d_9),(d_1,d_{10}),(d_2,d_{10}),(d_4,d_8)\}.\hfill \end{array}$$ Consequently, $$\begin{array}{cc}2\sum _{(d_i,d_j)\in A}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill & =2[8+8+8+4+4+2+2]=72\hfill \\ 2\sum _{(d_i,d_j)\in B}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill & =2[72+72+48+24+24+36+12+12+12+32+32+16+16+8+8+4]=856\hfill \\ 3\sum _{(d_i,d_j)\in C}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill & =3[24+12+6+16]=174\hfill \end{array}$$ Hence, the Wiener index of $\Gamma \prime ({\mathbb{Z}}_{72})$ is given by $$\begin{array}{cc}W(\Gamma ({\mathbb{Z}}_{72}))\hfill & =\frac{1}{2}\left[\sum _{i=1}^{\tau (n)-2}2\left|{\mathcal{A}}_{d_i}\right|\left(\left|{\mathcal{A}}_{d_j}\right|-1\right)+\sum _{\begin{array}{c}{d}_i\nmid d_j\\ d_j\nmid d_i\end{array}}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\right]+2\sum _{(d_i,d_j)\in A}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|+2\sum _{(d_i,d_j)\in B}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill \\ \hfill & +3\sum _{(d_i,d_j)\in C}\left|{\mathcal{A}}_{d_i}\right|\left|{\mathcal{A}}_{d_j}\right|\hfill \\ \hfill & =278+420+72+856+174=1800.\hfill \end{array}$$

Acknowledgments

The authors are thankful to the referee for valuable suggestions which helped in improving the presentation of the paper.

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There is no conflict of interest regarding the publishing of this paper.

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Funding

The second author gratefully acknowledge for providing financial support to CSIR (09/719(0093)/2019-EMR-I)) government of India.