On D_α-Spectrum of the Weakly Zero-Divisor Graph of ℤ_n

Abstract

Let us consider the finite commutative ring R, whose unity is $1\ne 0.$ Its weakly zero-divisor graph, represented as $W\Gamma \left(R\right)$, is a basic undirected graph with two distinct vertices, $c_1$ and $c_2$, that are adjacent if and only if there exist $r\in$ $\mathrm{ann}\left(c_1\right)$ and $s\in$ $\mathrm{ann}\left(c_2\right)$ that satisfy the condition $rs=0$. Let $D\left(G\right)$ be the distance matrix and $Tr\left(G\right)$ be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The $D_{\alpha }$ matrix of graph G is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$ for $\alpha \in [0,1]$. This article finds the $D_{\alpha }$ spectrum for the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ for various values of n and also shows that $W\Gamma \left({\mathbb{Z}}_n\right)$ for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes, is $D_{\alpha }$ integral.

1. Introduction

In this article, a commutative ring with the identity $1\ne 0$ shall be denoted by $R.$ When an element $c_2$, different from zero (0 $\ne c_2\in R$), exists such that $c_1{c}_2=0$, then the non-zero element $c_1$ is called a zero-divisor of R. $Z\left(R\right)$ is the collection of those zero-divisors in the ring R and ${Z\left(R\right)}^{*}=Z\left(R\right)\setminus \left\{0\right\}$.

The graph $G=(V,E)$ has been defined, where V denotes the set of vertices and E denotes the set of edges of G. When two distinct vertices of graph G, $c_1$ and $c_2$ are adjacent to each other in graph G, the notation $c_1\sim c_2$ represents this. In a graph G, the set of vertices adjacent to a vertex c is called its neighborhood; this neighborhood is represented by the notation $N_G\left(c\right)$. $K_m$ refers to the complete graph with m vertices and ${\overline{K}}_m$ refers to the complement of complete graph with m vertices. $deg\left(c\right)$, the degree of vertex c, represents the number of edges incident with $c\in V$. If $deg\left(c\right)=0$, then c is referred to as an isolated vertex. For every vertex $c\in G$, G is k-regular if $deg\left(c\right)=k$. The distance $D(v_i,v_j)$ between two vertices $v_i$ and $v_j$ is the shortest path from $v_i$ to $v_j$. Let A be any square matrix and let ${\lambda }_1,{\lambda }_2,{\lambda }_3,\dots ,{\lambda }_k$ be its different eigenvalues with multiplicities $f_1,f_2,f_3,\dots ,f_k$, respectively. The $spectrum$ of A is then denoted by $\sigma$(A), which is defined by

$$\sigma \left(A\right)=\left\{\begin{array}{ccccc}{\lambda }_1& {\lambda }_2& {\lambda }_3& \dots & {\lambda }_k\\ f_1& f_2& f_3& \dots & f_k\end{array}\right\}.$$

For a graph G, the $adjacency$ matrix $A\left(G\right)$ is a n-dimensional square matrix given by

$$A\left(G\right)=\left(a_{ij}\right)=\left\{\begin{array}{cc}1,\hfill & c_i\sim c_j\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

The convex linear combinations of the adjacency matrix $A\left(G\right)$ of G and the diagonal matrix of its vertices $D_{deg}\left(G\right)$ were proposed by Nikiforov [1]. This means that $A_{\alpha }\left(G\right)=\alpha A\left(G\right)+(1-\alpha )D_{deg}\left(G\right)$ for $\alpha \in [0,1],$ where $A_{\alpha }\left(G\right)$ is referred to as the generalized adjacency matrix or $A_{\alpha }\left(G\right)$ matrix of G. The vertex transmission in a graph is the sum of its distances to all other vertices in the graph G. The transmission matrix $T\left(G\right)$ of G is the diagonal matrix whose elements are the vertex transmissions and $D\left(G\right)$ is the distance matrix of graph G defined as

$$D\left(G\right)=\left\{\begin{array}{cc}D(v_i,v_j),\hfill & v_i\sim v_j\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

Similarly, the generalized distance matrix $D_{\alpha }\left(G\right)$ was presented by Cui, He, and Tian [2] as the convex combination of $T\left(G\right)$ and $D\left(G\right)$ for $\alpha \in [0,1],D_{\alpha }\left(G\right)=\alpha T\left(G\right)+(1-\alpha )D\left(G\right)$. If all the $D_{\alpha }$ eigenvalues of a graph G are integers, then G is said to be $D_{\alpha }$ integral.

Nikmehre et al. [3] introduced the idea of a weakly zero-divisor graph of ring R. The weakly zero-divisor graph of ring R is represented by the symbol $W\Gamma \left(R\right)$. There are two distinct vertices, $c_1$ and $c_2$, that are adjacent if and only if there exists $r\in$ $\mathrm{ann}\left(c_1\right)=\{c:c{c}_1=0\}$ and $s\in$ $\mathrm{ann}\left(c_2\right)=\{c:c{c}_2=0\}$, satisfying the condition $rs=0$. This undirected simple graph $W\Gamma \left(R\right)$ has a vertex set as the set of non-zero zero-divisors of R. Now, since the set of units of ${\mathbb{Z}}_n$ are $\varphi \left(n\right)$, this implies that the set of non-zero zero-divisors of ${\mathbb{Z}}_n$ are $n-\varphi \left(n\right)-1$. Hence, the order of $W\Gamma \left({\mathbb{Z}}_n\right)$ is $n-\varphi \left(n\right)-1$. The weakly zero-divisor graph’s spanning sub-graph is easily observed to be the zero-divisor graph of a ring.

In [4], the authors derived new properties on the $D_{\alpha }$ matrix including inequalities that involve the largest vertex transmission and the spectral radii of the distance matrix, distance signless Laplacian matrix, and $D_{\alpha }$ matrix. They also give a result for the spectrum of $D_{\alpha }$ matrix when G is the H-join of regular graphs.

Motivated by the results of [4], we have calculated the $D_{\alpha }$ spectrum of the weakly zero-divisor graph of ${\mathbb{Z}}_n$ for various values of n. More information about the spectra of graphs and additional information on various types of graphs based on a commutative ring can be found in [5,6,7,8,9,10,11]. The definitions, lemma’s, and theorems that are utilized to support the main results are analyzed in Section 2. $D_{\alpha }$ eigenvalues of $W\Gamma \left({\mathbb{Z}}_n\right)$ are looked into in Section 4, for $n={\vartheta }_1{\vartheta }_2,{{\vartheta }_1}^2{\vartheta }_2,{\vartheta }_1{\vartheta }_2{\vartheta }_3,{\vartheta }_1^{2j},{\vartheta }_1^{2j+1},{{\vartheta }_1}^t{\vartheta }_2$, where ${\vartheta }_1,{\vartheta }_2$, and ${\vartheta }_3$ are prime numbers with ${\vartheta }_1<{\vartheta }_2<{\vartheta }_3$, and $j\ge 3,t\ge 2$ being positive integers. Also, we calculate the $D_{\alpha }$ spectrum of the weakly zero-divisor $W\Gamma \left({\mathbb{Z}}_n\right)$, for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_s\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i^{\prime }$’s are distinct primes, and show that $W\Gamma \left({\mathbb{Z}}_n\right)$ is $D_{\alpha }$ integral.

2. Preliminaries

Definition 1.

Let $G(V,E)$ be a graph of order m with vertex set $\{c_1,c_2,\dots ,c_m\}$, and let $F_k(V_k,E_k)$ be disjoint graphs of order $m_k$ for $1\le k\le m.$ The generalized join graph $G[F_1,F_2,\dots ,F_m]$ is formed by taking the graphs $F_1,F_2,\dots ,F_m$ and, whenever vertices $c_k$ and $c_l$ are adjacent in G, joined each vertex of $F_k$ to every vertex of $F_l$.

$\tau \left(j_1\right)$ indicates the number of positive divisors of a positive integer $j_1$. For $j_2$ to not divide $j_1$, we write $j_2\nmid j_1$. The greatest common divisor of $j_1$ and $j_2$ is denoted by $(j_1,j_2).$ The number of positive integers smaller than or equal to $j_1$ that are relatively prime to $j_1$ is indicated by Euler’s phi function $\varphi \left(j_1\right)$. If $j_1={\vartheta }_1^{h_1}{\vartheta }_2^{h_2}\dots {\vartheta }_k^{h_k}$, where $h_1,h_2,\dots ,h_k$ are positive integers and ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_k$ are distinct primes, then $j_1$ is in $prime decomposition$.

Lemma 1

([12]). If $s_1={\xi }_1^{h_1}{\xi }_2^{h_2}\dots {\xi }_k^{h_k}$ is a prime decomposition of $s_1$, then $\tau \left(s_1\right)=(h_1+1)(h_2+1)\dots (h_k+1).$

Let $j_1,j_2,\dots ,j_k$ be the proper divisors of n. For $1\le i\le k,$ consider the following sets

$$A_{j_i}=\{x\in {\mathbb{Z}}_n:(x,n)=j_i\}.$$

Moreover, observe that for $i\ne s$, $A_{j_i}\cap A_{j_s}=\varnothing$. As a result, the vertex set of $W\Gamma \left({\mathbb{Z}}_n\right)$ has a partition formed by the sets $A_{j_1},A_{j_2},\dots ,A_{j_k}$. $V(W\Gamma \left({\mathbb{Z}}_n\right))=A_{j_1}\cup A_{j_2}\cup \dots \cup A_{j_k}$ as a result. The following lemma provides information about the cardinality of each $A_{j_i}$.

Lemma 2

([13] Proposition 2.1). Let $j_i$ be a proper divisor of n. Then, $\left|A_{j_i}\right|=\varphi \left({\displaystyle \frac{n}{j_i}}\right)$ for $1\le i\le k$.

Lemma 3

([14]). Let n be represented as $n=l_1{l}_2\dots l_m{w_1}^{s_1}{w_2}^{s_2}\dots {w_i}^{s_i}$, where $l_i\prime s, w_i\prime s$ are distinct primes and $i\ge 0, s_i\ge 2$ and $m\ge 1$. Then, consider the set of proper divisors of n, denoted as $\{j_1,j_2,\dots ,j_k\}$. If $j_r\in \{l_1,l_2,\dots ,l_m\}$, then the induced sub-graph of $W\Gamma \left({\mathbb{Z}}_n\right)$ by $A_{j_r}$ is ${\overline{K}}_{\varphi \left({\displaystyle \frac{n}{j_r}}\right)}.$

Corollary 1

([14]). Let $j_t$ be the proper divisor of positive integer $n.$ The following assertions are true:

(1)

For $t\in \{1,2,\dots ,s\}$, the induced subgraph $W\Gamma \left(A_{j_t}\right)$ of $W\Gamma \left({\mathbb{Z}}_n\right)$, formed by the vertices in the set $A_{j_t}$, takes one of two forms: either ${\overline{K}}_{\varphi \left({\displaystyle \frac{n}{j_t}}\right)}$ or $K_{\varphi \left({\displaystyle \frac{n}{j_t}}\right)}$.

(2)

For $t,q\in \{1,2,\dots ,s\}$ and $t\ne q$, a vertex within $A_{j_t}$ is connected to either all or none of the vertices in $A_{j_q}$ in the graph $W\Gamma \left({\mathbb{Z}}_n\right).$

The sub-graphs $W\Gamma \left(A_{j_t}\right)$ created within the structure of $W\Gamma \left({\mathbb{Z}}_n\right)$ can be classified as either complete graphs or empty graphs, as shown by the previously noted Corollary 1. Let the distinct proper divisors of n be $j_1,j_2,\dots ,j_s$, then the vertex set of simple graph ${\delta }_n^{*}$ is $\{j_1,j_2,\dots ,j_s\}$ and $j_i\sim j_k$ for all $i=k$.

Lemma 4

([14]). $W\Gamma \left({\mathbb{Z}}_n\right)={\delta }_n^{*}[W\Gamma \left(A_{j_1}\right),W\Gamma \left(A_{j_2}\right),\dots ,W\Gamma \left(A_{j_s}\right)]$ where $j_1,j_2,\dots ,j_s$ are all the proper divisors of n.

The following theorem provides the generalized join graph’s $D_{\alpha }$ spectrum in terms of the spectrum of the adjacency matrix of regular graphs.

Theorem 1

([4]). Let H be a connected graph of order $s.$ Let $\alpha \in [0,1].$ If, for $i=1,2,\dots ,s$, $G_i$ is a $r_i$-regular graph of order $m_i,$ then the $D_{\alpha }$ spectrum of the H-join graphs $G_1,G_2,\dots ,G_s$ is

$$\sigma \left(D_{\alpha }\left(G\right)\right)=\bigcup _{i=1}^s(\sigma \left(M_i\left(\alpha \right)\right)-\left\{{\mu }_i\left(\alpha \right)\right\})\bigcup \sigma \left(Y\right),$$

where

$$Y=\left[\begin{array}{cccc}{\mu }_1\left(\alpha \right)& \delta d_{1,2}\sqrt{m_1{m}_2}& \dots & \delta d_{1,s}\sqrt{m_1{m}_s}\\ \delta d_{2,1}\sqrt{m_2{m}_1}& {\mu }_2\left(\alpha \right)& \dots & \delta d_{2,s}\sqrt{m_2{m}_s}\\ ⋮& ⋮& \ddots & ⋮\\ \delta d_{s,1}\sqrt{m_s{m}_1}& \delta d_{s,2}\sqrt{m_s{m}_2}& \dots & {\mu }_s\left(\alpha \right)\end{array}\right].$$

(1)

Here, $\delta =(1-\alpha ),\sigma \left(M_i\left(\alpha \right)\right)=\alpha Tr\left(v\right)+(1-\alpha )(2(m_i-1)-r_i),\alpha Tr\left(v\right)-(1-\alpha )(2+{\lambda }_{m_i}\left(A\left(G_i\right)\right)),\dots ,\alpha Tr\left(v\right)-(1-\alpha )(2+{\lambda }_2\left(A\left(G_i\right)\right)),{\mu }_i\left(\alpha \right)=\alpha Tr\left(v\right)+(1-\alpha )(2(m_i-1)-r_i),Tr\left(v\right)=2(m_i-1)-r_i+{\displaystyle \sum _{j=1j\ne i}^s}{m}_j{d}_{i,j}$ and $d_{i,j}$ are the distance from vertex i to j for $1\le i, j\le s.$

3. Methodology

Research in graph theory continues to flourish because it provides a link between discrete structures and pure as well as applied mathematics. Using sophisticated mathematical tools, the study’s method builds upon well-established ideas in algebra and graph theory to produce new results. Our efforts rely on using the content of existing research to expand on established findings and investigate fresh aspects of weakly zero-divisor graphs.

The analysis in this paper heavily relies on the use of matrix theory and linear algebra. In particular, spectral graph theory provides a strong framework for studying the interaction between algebraic and graph-theoretical characteristics, a crucial tool for capturing the structural features of the weakly zero-divisor graph of the ring ${\mathbb{Z}}_n.$

The primary objective of this study is to analyze the $D_{\alpha }$ spectra of the weakly zero-divisor graph $W\Gamma \left({\mathbb{Z}}_n\right)$ for a general class of n, $n={\vartheta }_1{\vartheta }_2\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2, t\ge 1,$$s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes. To achieve this, we use the concept of new results on the $D_{\alpha }$-matrix of connected graphs, which was introduced by Diaz et al. [4].

4. Results

We will prove the main results of this paper in this section. For $r\in \{1,2,\dots ,k\}$, the induced subgraph $W\Gamma \left(A_{j_r}\right)$ of $W\Gamma \left({\mathbb{Z}}_n\right)$, formed by the vertices in the set $A_{j_r}$, is either ${\overline{K}}_{\varphi \left({\displaystyle \frac{n}{j_r}}\right)}$ or $K_{\varphi \left({\displaystyle \frac{n}{j_r}}\right)}.$ Recall that the adjacency spectrum of complete graph $K_l$ and its complement graph $\overline{K_l}$ on l vertices is given by

$$\sigma \left(K_l\right)=\left\{\begin{array}{cc}l-1& -1\\ 1& l-1\end{array}\right\}\mathrm{and}\sigma \left(\overline{K_l}\right)=\left\{\begin{array}{c}0\\ l\end{array}\right\}, \mathrm{respectively}.$$

Lemma 5.

Let n be the product of two different primes ${\vartheta }_1$ and ${\vartheta }_2$. Then, the graph $W\Gamma \left({\mathbb{Z}}_n\right)$’s $D_{\alpha }$ spectrum is given by

$$\sigma \left(D_{\alpha }(W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2}\right))\right)=\left\{\begin{array}{cc}2\alpha {\vartheta }_2+\alpha {\vartheta }_1-3\alpha -2& 2\alpha {\vartheta }_1+\alpha {\vartheta }_2-3\alpha -2\\ {\vartheta }_2-2& {\vartheta }_1-2\end{array}\right\}.$$

The remaining two $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ are the roots of the characteristic polynomial ${\lambda }^2-\lambda (\alpha {\vartheta }_1+\alpha {\vartheta }_2+2{\vartheta }_1+2{\vartheta }_2-2\alpha -8)+{\alpha }^2{\vartheta }_1{\vartheta }_2-{\alpha }^2{\vartheta }_1-{\alpha }^2{\vartheta }_2+{\alpha }^2+2\alpha {{\vartheta }_1}^2-6\alpha {\vartheta }_1+2\alpha {{\vartheta }_2}^2-6\alpha {\vartheta }_2+8\alpha -{\delta }^2{\vartheta }_1{\vartheta }_2+{\delta }^2{\vartheta }_1+{\delta }^2{\vartheta }_2-{\delta }^2+4{\vartheta }_1{\vartheta }_2-8{\vartheta }_1-8{\vartheta }_2+16=0.$

Proof. 

The proper divisors of n are ${\vartheta }_1$ and ${\vartheta }_2$ and ${\vartheta }_1<{\vartheta }_2$. Also, by the definition of ${\delta }_n^{*};{\vartheta }_1\sim {\vartheta }_2.$ Now, by Lemma 4, we have $W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2}\right)={\delta }_{{\vartheta }_1{\vartheta }_2}^{*}[W\Gamma \left(A_{{\vartheta }_1}\right),W\Gamma \left(A_{{\vartheta }_2}\right)]$. Therefore, by Lemma 2 and Corollary 1, we have $W\Gamma \left(A_{{\vartheta }_1}\right)={\overline{K}}_{\varphi \left({\vartheta }_2\right)}$ and $W\Gamma \left(A_{{\vartheta }_2}\right)={\overline{K}}_{\varphi \left({\vartheta }_1\right)}$. Therefore, by Theorem 1, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2}\right)$ is

$$\left\{\begin{array}{cc}2\alpha {\vartheta }_2+\alpha {\vartheta }_1-3\alpha -2& 2\alpha {\vartheta }_1+\alpha {\vartheta }_2-3\alpha -2\\ {\vartheta }_2-2& {\vartheta }_1-2\end{array}\right\}\bigcup {\sigma }_{\alpha }\left(Y\right)$$

and the root of characteristic polynomial ${\lambda }^2-\lambda (\alpha {\vartheta }_1+\alpha {\vartheta }_2+2{\vartheta }_1+2{\vartheta }_2-2\alpha -8)+{\alpha }^2{\vartheta }_1{\vartheta }_2-{\alpha }^2{\vartheta }_1-{\alpha }^2{\vartheta }_2+{\alpha }^2+2\alpha {{\vartheta }_1}^2-6\alpha {\vartheta }_1+2\alpha {{\vartheta }_2}^2-6\alpha {\vartheta }_2+8\alpha -{\delta }^2{\vartheta }_1{\vartheta }_2+{\delta }^2{\vartheta }_1+{\delta }^2{\vartheta }_2-{\delta }^2+4{\vartheta }_1{\vartheta }_2-8{\vartheta }_1-8{\vartheta }_2+16=0$ of the matrix provided below can be used to determine the remaining two eigenvalues

$$Y=\left[\begin{array}{cc}\alpha {\vartheta }_1+2{\vartheta }_2-\alpha -4& \delta \sqrt{({\vartheta }_1-1)({\vartheta }_2-1)}\\ \delta \sqrt{({\vartheta }_1-1)({\vartheta }_2-1)}& \alpha {\vartheta }_2+2{\vartheta }_1-\alpha -4\end{array}\right]$$

(2)

where $\delta =(1-\alpha )$. □

Theorem 2.

For distinct prime ${\vartheta }_1,{\vartheta }_2,{\vartheta }_3$ and $n={\vartheta }_1{\vartheta }_2{\vartheta }_3$, the $D_{\alpha }$ spectrum of the $W\Gamma \left({\mathbb{Z}}_n\right)$ is

$$\left\{\begin{array}{cccc}\alpha \left|V\right|-1& \alpha \left(\varphi \left({\vartheta }_2{\vartheta }_3\right)\right)+\alpha \left|V\right|-2& \alpha \left(\varphi \left({\vartheta }_1{\vartheta }_3\right)\right)+\alpha \left|V\right|-2& A\\ \varphi \left({\vartheta }_1\right)+\varphi \left({\vartheta }_2\right)+\varphi \left({\vartheta }_3\right)-3& \varphi \left({\vartheta }_2{\vartheta }_3\right)-1& \varphi \left({\vartheta }_1{\vartheta }_3\right)-1& \varphi \left({\vartheta }_1{\vartheta }_2\right)-1\end{array}\right\}$$

where $A=\alpha \left(\varphi \left({\vartheta }_1{\vartheta }_2\right)\right)+\alpha \left|V\right|-2$ and the cardinality $\left|V\right|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2{\vartheta }_3}\right)$ is given $\left|V\right|=\varphi \left({\vartheta }_1\right)+\varphi \left({\vartheta }_2\right)+\varphi \left({\vartheta }_3\right)+\varphi \left({\vartheta }_1{\vartheta }_2\right)+\varphi \left({\vartheta }_1{\vartheta }_3\right)+\varphi \left({\vartheta }_2{\vartheta }_3\right).$ The remaining six $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ are the eigenvalues of matrix (3).

Proof. 

Let $n={\vartheta }_1{\vartheta }_2{\vartheta }_3$, where ${\vartheta }_1<{\vartheta }_2<{\vartheta }_3$, note that ${\delta }_{{\vartheta }_1{\vartheta }_2{\vartheta }_3}^{*}$ is a complete graph on vertices $\{{\vartheta }_1,{\vartheta }_2,{\vartheta }_3,{\vartheta }_1{\vartheta }_2,{\vartheta }_1{\vartheta }_3,{\vartheta }_2{\vartheta }_3\}$. Now, by Lemma 4, we have

$$W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2{\vartheta }_3}\right)={\delta }_{{\vartheta }_1{\vartheta }_2{\vartheta }_3}^{*}[W\Gamma \left(A_{{\vartheta }_1}\right),W\Gamma \left(A_{{\vartheta }_2}\right),W\Gamma \left(A_{{\vartheta }_3}\right),W\Gamma \left(A_{{\vartheta }_1{\vartheta }_2}\right),W\Gamma \left(A_{{\vartheta }_1{\vartheta }_3}\right),W\Gamma \left(A_{{\vartheta }_2{\vartheta }_3}\right)].$$

Therefore, by Lemma 2 and Corollary 1, we have $W\Gamma \left(A_{{\vartheta }_1}\right)={\overline{K}}_{\varphi \left({\vartheta }_2{\vartheta }_3\right)},W\Gamma \left(A_{{\vartheta }_2}\right)={\overline{K}}_{\varphi \left({\vartheta }_1{\vartheta }_3\right)},W\Gamma \left(A_{{\vartheta }_3}\right)={\overline{K}}_{\varphi \left({\vartheta }_1{\vartheta }_2\right)},W\Gamma \left(A_{{\vartheta }_1{\vartheta }_3}\right)=K_{\varphi \left({\vartheta }_2\right)}$ and $W\Gamma \left(A_{{\vartheta }_2{\vartheta }_3}\right)=K_{\varphi \left({\vartheta }_1\right)}$.

• The cardinality $\left|V\right|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2{\vartheta }_3}\right)$ is given by $\varphi \left({\vartheta }_1\right)+\varphi \left({\vartheta }_2\right)+\varphi \left({\vartheta }_3\right)+\varphi \left({\vartheta }_1{\vartheta }_2\right)+\varphi \left({\vartheta }_1{\vartheta }_3\right)+\varphi \left({\vartheta }_2{\vartheta }_3\right).$ Also, we have $m_1=\varphi \left({\vartheta }_2{\vartheta }_3\right),m_2=\varphi \left({\vartheta }_1{\vartheta }_3\right),m_3=\varphi \left({\vartheta }_1{\vartheta }_2\right),m_4=\varphi \left({\vartheta }_3\right),m_5=\varphi \left({\vartheta }_2\right)$ and $m_6=\varphi \left({\vartheta }_1\right).$ It follows that $r_i=0,r_4=\varphi \left({\vartheta }_3\right)-1,r_5=\varphi \left({\vartheta }_2\right)-1$ and $r_6=\varphi \left({\vartheta }_1\right)-1$ for $1\le i\le 3.$ Therefore, by Theorem 1, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{\vartheta }_1{\vartheta }_2{\vartheta }_3}\right)$ is

$$\left\{\begin{array}{cccc}\alpha \left|V\right|-1& \alpha \left(\varphi \left({\vartheta }_2{\vartheta }_3\right)\right)+\alpha \left|V\right|-2& \alpha \left(\varphi \left({\vartheta }_1{\vartheta }_3\right)\right)+\alpha \left|V\right|-2& A\\ \varphi \left({\vartheta }_1\right)+\varphi \left({\vartheta }_2\right)+\varphi \left({\vartheta }_3\right)-3& \varphi \left({\vartheta }_2{\vartheta }_3\right)-1& \varphi \left({\vartheta }_1{\vartheta }_3\right)-1& \varphi \left({\vartheta }_1{\vartheta }_2\right)-1\end{array}\right\}$$

where $A=\alpha \left(\varphi \left({\vartheta }_1{\vartheta }_2\right)\right)+\alpha \left|V\right|-2.$ The matrix’s characteristic polynomial can be used to determine the remaining six eigenvalues,

$$Y=\left[\begin{array}{cccccc}A″& \delta \sqrt{\beta \gamma }& \delta \sqrt{\xi \beta }& \delta \sqrt{\beta C}& \delta \sqrt{\beta B}& \delta \sqrt{\beta A}\\ \delta \sqrt{\gamma \beta }& B″& \delta \sqrt{\gamma \xi }& \delta \sqrt{\gamma C}& \delta \sqrt{\gamma B}& \delta \sqrt{\gamma A}\\ \delta \sqrt{\xi \beta }& \delta \sqrt{\xi \gamma }& C″& \delta \sqrt{\xi C}& \delta \sqrt{\xi B}& \delta \sqrt{\xi A}\\ \delta \sqrt{C\beta }& \delta \sqrt{C\gamma }& \delta \sqrt{C\xi }& D″& \delta \sqrt{CB}& \delta \sqrt{AC}\\ \delta \sqrt{B\beta }& \delta \sqrt{B\gamma }& \delta \sqrt{B\xi }& \delta \sqrt{BC}& E″& \delta \sqrt{AB}\\ \delta \sqrt{A\beta }& \delta \sqrt{A\gamma }& \delta \sqrt{A\xi }& \delta \sqrt{AC}& \delta \sqrt{AB}& F″\end{array}\right]$$

(3)

where $A″=\alpha \left|V\right|-\varphi \left({\vartheta }_2{\vartheta }_3\right)(\alpha -2)-2,B″=\alpha \left|V\right|-\varphi \left({\vartheta }_1{\vartheta }_3\right)(\alpha -2)-2,C″=\alpha \left|V\right|-\varphi \left({\vartheta }_1{\vartheta }_2\right)(\alpha -2)-2,D″=\alpha \left|V\right|-\varphi \left({\vartheta }_3\right)(\alpha -1)-1,E″=\alpha \left|V\right|-\varphi \left({\vartheta }_2\right)(\alpha -1)-1,F″=\alpha \left|V\right|-\varphi \left({\vartheta }_1\right)(\alpha -1)-1,\xi =\varphi \left({\vartheta }_1{\vartheta }_2\right),\beta =\varphi \left({\vartheta }_2{\vartheta }_3\right),\gamma =\varphi \left({\vartheta }_1{\vartheta }_3\right),C=\varphi \left({\vartheta }_3\right),B=\varphi \left({\vartheta }_2\right),A=\varphi \left({\vartheta }_1\right)$ and $\delta =(1-\alpha ).$ □

Theorem 3.

Let $n={\vartheta }_1^K$ where $K=2j$, ${\vartheta }_1$ is a prime, and $j\ge 3$ is a positive integer. Then, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j}}\right)$ consists of eigenvalue $\alpha \left|V\right|-1$ with multiplicity ${\vartheta }_1^{2j-1}-2j$, where $\left|V\right|={\displaystyle \sum _{i=1}^{2j-1}\varphi \left({\vartheta }_1^i\right)}.$ The other remaining $2j-1$, $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j}}\right)$ are eigenvalues of matrix (4).

Proof. 

For $n={\vartheta }_1^{2j}$, where j is a positive integer and ${\vartheta }_1$ is a prime, the proper divisors of ${\vartheta }_1^{2j}$ are ${\vartheta }_1,{\vartheta }_1^2,{\vartheta }_1^3,\dots ,{\vartheta }_1^{j-1},{\vartheta }_1^j,{\vartheta }_1^{j+1},\dots ,{\vartheta }_1^{2j-2},{\vartheta }_1^{2j-1}.$ By Lemma 4, we have

$$W\Gamma \left({\mathbb{Z}}_{{\vartheta }^{2j}}\right)={\delta }_{{{\vartheta }_1}^{2j}}^{*}[W\Gamma \left(A_{{\vartheta }_1}\right),W\Gamma \left(A_{{{\vartheta }_1}^2}\right),\dots ,W\Gamma \left(A_{{{\vartheta }_1}^j}\right),\dots ,W\Gamma \left(A_{{{\vartheta }_1}^{2j-2}}\right),W\Gamma \left(A_{{{\vartheta }_1}^{2j-1}}\right)].$$

It follows that $\left|V\right|={\displaystyle \sum _{i=1}^{2j-1}\varphi \left({\vartheta }_1^i\right)}$, where $\left|V\right|$ is the cardinality of vertex set V of $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j}}\right)$. Therefore, by Lemma 2 and Corollary 1, we get

$$\begin{array}{c}\hfill W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j}}\right)={\delta }_{{{\vartheta }_1}^{2j}}^{*}[K_{\varphi \left({{\vartheta }_1}^{2j-1}\right)},K_{\varphi \left({{\vartheta }_1}^{2j-2}\right)},\dots ,,K_{\varphi \left({{\vartheta }_1}^{j+1}\right)},K_{\varphi \left({{\vartheta }_1}^j\right)},\dots ,K_{\varphi \left({{\vartheta }_1}^2\right)},K_{\varphi \left({\vartheta }_1\right)}].\end{array}$$

$m_i=\varphi \left({{\vartheta }_1}^{2j-i}\right)$ and $r_i=\varphi \left({{\vartheta }_1}^{2j-i}\right)-1$ for $i=1,2,3,\dots ,2j-2,2j-1.$ Therefore, by Theorem 1, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j}}\right)$ consists of eigenvalue $\alpha \left|V\right|-1$ with multiplicity ${\vartheta }_1^{2j-1}-2j.$ And the roots of matrix (4)’s characteristic polynomial can be used to determine the remained $2j-1$ eigenvalues,

$$\left[\begin{array}{ccccccc}P& \delta \sqrt{P″}& \dots & \delta \sqrt{Q″}& \dots & \delta \sqrt{R″}& \delta \sqrt{S″}\\ \delta \sqrt{P″}& Q& \dots & \delta \sqrt{T″}& \dots & \delta \sqrt{U″}& \delta \sqrt{V″}\\ ⋮& ⋮& \ddots & ⋮& \dots & ⋮& ⋮\\ \delta \sqrt{Q″}& \delta \sqrt{T″}& \dots & R& \dots & \delta \sqrt{W″}& \delta \sqrt{X″}\\ ⋮& ⋮& \dots & ⋮& \ddots & ⋮& ⋮\\ \delta \sqrt{R″}& \delta \sqrt{U″}& \dots & \delta \sqrt{W″}& \dots & S& \delta \sqrt{Y″}\\ \delta \sqrt{S″}& \delta \sqrt{V″}& \dots & \delta \sqrt{X″}& \dots & \delta \sqrt{Y″}& T\end{array}\right].$$

(4)

where $P=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^{2j-1}\right)(\alpha -1)-1,Q=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^{2j-2}\right)(\alpha -1)-1,R=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^j\right)(\alpha -1)-1,S=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^2\right)(\alpha -1)-1,T=\alpha \left|V\right|-\varphi \left({\vartheta }_1\right)(\alpha -1)-1,P″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({{\vartheta }_1}^{2j-2}\right),Q″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({{\vartheta }_1}^j\right),R″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({{\vartheta }_1}^2\right),S″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({\vartheta }_1\right),T″=\varphi \left({{\vartheta }_1}^{2j-2}\right)\varphi \left({{\vartheta }_1}^j\right),U″=\varphi \left({{\vartheta }_1}^{2j-2}\right)\varphi \left({{\vartheta }_1}^2\right),V″=\varphi \left({{\vartheta }_1}^{2j-2}\right)\varphi \left({\vartheta }_1\right),W″=\varphi \left({{\vartheta }_1}^j\right)\varphi \left({{\vartheta }_1}^2\right),X″=\varphi \left({{\vartheta }_1}^j\right)\varphi \left({\vartheta }_1\right),Y″=\varphi \left({{\vartheta }_1}^2\right)\varphi \left({\vartheta }_1\right),\left|V\right|={\displaystyle \sum _{i=1}^{2j-1}}\varphi \left({\vartheta }_1^i\right)$ and $\delta =1-\alpha .$ □

Theorem 4.

Let $n={\vartheta }_1^K$ where $K=2j+1,$ ${\vartheta }_1$ a prime, and $j\ge 3$ is a positive integer. Then, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j+1}}\right)$ consists of eigenvalue $\alpha \left|V\right|-1$ with multiplicity ${\vartheta }_1^{2j}-(2j+1)$. The other remaining $2j,$ $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j+1}}\right)$ are eigenvalues of matrix (5).

Proof. 

Similarly, as above for Theorem 3, we can prove that the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j+1}}\right)$ consists of eigenvalue $\alpha \left|V\right|-1$ with multiplicities ${\vartheta }_1^{2j}-(2j+1)$, $\left|V\right|={\displaystyle \sum _{i=1}^{2j}\varphi \left({\vartheta }_1^i\right)},$ where $\left|V\right|$ is the cardinality of vertex set V of $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j+1}}\right)$.

$$\left[\begin{array}{ccccccc}P\prime & \delta \sqrt{A″}& \dots & \delta \sqrt{B″}& \dots & \delta \sqrt{C″}& \delta \sqrt{D″}\\ \delta \sqrt{A″}& Q\prime & \dots & \delta \sqrt{E″}& \dots & \delta \sqrt{F″}& \delta \sqrt{G″}\\ ⋮& ⋮& \ddots & ⋮& \dots & ⋮& ⋮\\ \delta \sqrt{B″}& \delta \sqrt{E″}& \dots & R\prime & \dots & \delta \sqrt{H″}& \delta \sqrt{I″}\\ ⋮& ⋮& \dots & ⋮& \ddots & ⋮& ⋮\\ \delta \sqrt{C″}& \delta \sqrt{F″}& \dots & \delta \sqrt{H″}& \dots & S\prime & \delta \sqrt{K″}\\ \delta \sqrt{D″}& \delta \sqrt{G″}& \dots & \delta \sqrt{I″}& \dots & \delta \sqrt{K″}& T\prime \end{array}\right].$$

(5)

The other remaining $2j$, $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^{2j+1}}\right)$ are eigenvalues of matrix (5), which is obtained from matrix (1), where $P\prime =\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^{2j}\right)(\alpha -1)-1,Q\prime =\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^{2j-1}\right)(\alpha -1)-1,R\prime =\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^j\right)(\alpha -1)-1,S\prime =\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^2\right)(\alpha -1)-1,T\prime =\alpha \left|V\right|-\varphi \left({\vartheta }_1\right)(\alpha -1)-1,A″=\varphi \left({{\vartheta }_1}^{2j}\right)\varphi \left({{\vartheta }_1}^{2j-1}\right),B″=\varphi \left({{\vartheta }_1}^{2j}\right)\varphi \left({{\vartheta }_1}^j\right),C″=\varphi \left({{\vartheta }_1}^{2j}\right)\varphi \left({{\vartheta }_1}^2\right),D″=\varphi \left({{\vartheta }_1}^{2j}\right)\varphi \left({\vartheta }_1\right),E″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({{\vartheta }_1}^j\right),F″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({{\vartheta }_1}^2\right),G″=\varphi \left({{\vartheta }_1}^{2j-1}\right)\varphi \left({\vartheta }_1\right),H″=\varphi \left({{\vartheta }_1}^j\right)\varphi \left({{\vartheta }_1}^2\right),I″=\varphi \left({{\vartheta }_1}^j\right)\varphi \left({\vartheta }_1\right),K″=\varphi \left({{\vartheta }_1}^2\right)\varphi \left({\vartheta }_1\right),\left|V\right|={\displaystyle \sum _{i=1}^{2j}\varphi \left({\vartheta }_1^i\right)}$ and $\delta =1-\alpha$. □

If $j=3$ in Theorem 3, the resulting outcome gives the $D_{\alpha }$ spectrum of $W\Gamma \left({\mathbb{Z}}_n\right)$.

Corollary 2.

The $D_{\alpha }$ spectrum of $W\Gamma \left({\mathbb{Z}}_n\right)$ for $n={{\vartheta }_1}^6$ consists of eigenvalue $\alpha ({\vartheta }_1^5-1)-1$ with multiplicity ${\vartheta }_1^5-6$. The remaining 5, $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^6}\right)$ are eigenvalues of matrix (6),

$$\left[\begin{array}{ccccc}{\mu }_1\left(\alpha \right)& \delta {\vartheta }_1^3\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& \delta {\vartheta }_1^3\left({\vartheta }_1-1\right)& \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)\\ \delta {\vartheta }_1^3\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& {\mu }_2\left(\alpha \right)& \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)& \delta {\vartheta }_1\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}\\ \delta {\vartheta }_1^3\left({\vartheta }_1-1\right)& \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& {\mu }_3\left(\alpha \right)& \delta {\vartheta }_1\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& \delta {\vartheta }_1\left({\vartheta }_1-1\right)\\ \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)& \delta {\vartheta }_1\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& {\mu }_4\left(\alpha \right)& \delta \left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}\\ \delta {\vartheta }_1^2\left({\vartheta }_1-1\right)& \delta {\vartheta }_1\left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& \delta {\vartheta }_1\left({\vartheta }_1-1\right)& \delta \left({\vartheta }_1-1\right)\sqrt{{\vartheta }_1}& {\mu }_5\left(\alpha \right)\end{array}\right].$$

(6)

where ${\mu }_1\left(\alpha \right)={\vartheta }_1^4(\alpha -1)+{\vartheta }_1^5-(\alpha +1),{\mu }_2\left(\alpha \right)=\alpha {\vartheta }_1^5+{\vartheta }_1^4(1-\alpha )+{\vartheta }_1^3(\alpha -1)-(\alpha +1),m{u}_3\left(\alpha \right)=\alpha {\vartheta }_1^5+{\vartheta }_1^3(1-\alpha )+{\vartheta }_1^2(\alpha -1)-(\alpha +1),{\mu }_4\left(\alpha \right)=\alpha {\vartheta }_1^5+{\vartheta }_1^2(1-\alpha )+{\vartheta }_1(\alpha -1)-(\alpha +1),{\mu }_5\left(\alpha \right)=\alpha {\vartheta }_1^5+{\vartheta }_1(1-\alpha )-2$ and $\delta =1-\alpha .$

Theorem 5.

For distinct primes ${\vartheta }_1,{\vartheta }_2$ and $n={{\vartheta }_1}^t{\vartheta }_2$, $t\ge 2$. The $D_{\alpha }$ spectrum of the $W\Gamma \left({\mathbb{Z}}_n\right)$ consists of eigenvalues

$$\left\{\begin{array}{cc}\alpha \left|V\right|-1& \alpha \left(\varphi \left({\vartheta }_1^t\right)\right)+\alpha \left|V\right|-2\\ {\displaystyle \sum _{k=1}^t\varphi \left({{\vartheta }_1}^{t-k}{\vartheta }_2\right)+\sum _{k=1}^{t-1}\varphi \left({{\vartheta }_1}^k\right)-(2t-1)}& \varphi \left({{\vartheta }_1}^t\right)-1\end{array}\right\}.$$

The cardinality $\left|V\right|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^t{\vartheta }_2}\right)$ is given by ${\displaystyle \sum _{k=1}^t\varphi \left({\vartheta }_1^k\right)+\sum _{k=0}^{t-1}\varphi \left({{\vartheta }_1}^k{\vartheta }_2\right)}$ and the roots of the characteristic polynomial of matrix (7) provide the remaining $2t$ eigenvalues.

Proof. 

Let $n={{\vartheta }_1}^t{\vartheta }_2$, where ${\vartheta }_1<{\vartheta }_2$, note that ${\delta }_{{{\vartheta }_1}^t{\vartheta }_2}^{*}$ is complete graph on vertices $\{{\vartheta }_1,{\vartheta }_1^2,\dots ,{{\vartheta }_1}^t,{\vartheta }_2,{\vartheta }_1{\vartheta }_2,{{\vartheta }_1}^2{\vartheta }_2,\dots ,{{\vartheta }_1}^{t-1}{\vartheta }_2\}$. By Lemma 4, we have

$$\begin{array}{cc}\hfill W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^t{\vartheta }_2}\right)={\delta }_{{{\vartheta }_1}^t{\vartheta }_2}^{*}& [W\Gamma \left(A_{{\vartheta }_1}\right),\dots ,W\Gamma \left(A_{{{\vartheta }_1}^t}\right),W\Gamma \left(A_{{\vartheta }_2}\right),W\Gamma \left(A_{{\vartheta }_1{\vartheta }_2}\right),,\dots ,W\Gamma \left(A_{{{\vartheta }_1}^{t-1}{\vartheta }_2}\right)].\hfill \end{array}$$

Therefore, by Lemma 2 and Corollary 1, we get

$$\begin{array}{cc}\hfill W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^t{\vartheta }_2}\right)={\delta }_{{{\vartheta }_1}^t{\vartheta }_2}^{*}& [K_{\varphi \left({{\vartheta }_1}^{t-1}{\vartheta }_2\right)},\dots ,K_{\varphi \left({\vartheta }_2\right)},{\overline{K}}_{\varphi \left({{\vartheta }_1}^t\right)},K_{\varphi \left({{\vartheta }_1}^{t-1}\right)},\dots ,K_{\varphi \left({\vartheta }_1\right)}].\hfill \end{array}$$

Consequently, the cardinality $\left|V\right|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^t{\vartheta }_2}\right)$ is given by ${\displaystyle \sum _{k=1}^t\varphi \left({\vartheta }_1^k\right)+\sum _{k=0}^{t-1}\varphi \left({{\vartheta }_1}^k{\vartheta }_2\right)}$ and also $m_{{\vartheta }_1}=\varphi \left({{\vartheta }_1}^{t-1}{\vartheta }_2\right),m_{{{\vartheta }_1}^2}=\varphi \left({{\vartheta }_1}^{t-2}{\vartheta }_2\right),\dots ,m_{{{\vartheta }_1}^t}=\varphi \left({\vartheta }_2\right),m_{{\vartheta }_1{\vartheta }_2}=\varphi \left({{\vartheta }_1}^{t-1}\right),\dots ,m_{{{\vartheta }_1}^r{\vartheta }_2}=\varphi \left({{\vartheta }_1}^{t-r}\right),\dots ,m_{{{\vartheta }_1}^{t-1}{\vartheta }_2}=\varphi \left({\vartheta }_1\right)$ and $m_{{\vartheta }_2}=\varphi \left({{\vartheta }_1}^t\right).$ And it follows that $r_{{\vartheta }_1}=\varphi \left({{\vartheta }_1}^{t-1}{\vartheta }_2\right)-1,r_{{{\vartheta }_1}^2}=\varphi \left({{\vartheta }_1}^{t-2}{\vartheta }_2\right)-1,\dots ,r_{{{\vartheta }_1}^t}=\varphi \left({\vartheta }_2\right)-1,r_{{\vartheta }_1{\vartheta }_2}=\varphi \left({{\vartheta }_1}^{t-1}\right)-1,\dots ,r_{{{\vartheta }_1}^r{\vartheta }_2}=\varphi \left({{\vartheta }_1}^{t-r}\right)-1,\dots ,r_{{{\vartheta }_1}^{t-1}{\vartheta }_2}=\varphi \left({\vartheta }_1\right)-1$ and $r_{{\vartheta }_2}=0.$ Therefore, by Theorem 1, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^t{\vartheta }_2}\right)$ is

$$\begin{array}{c}\hfill \sigma \left(D_{\alpha }(W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^t{\vartheta }_2}\right))\right)=\left\{\begin{array}{cc}\alpha \left|V\right|-1& \alpha \left(\varphi \left({\vartheta }_1^t\right)\right)+\alpha \left|V\right|-2\\ {\displaystyle \sum _{k=1}^t\varphi \left({{\vartheta }_1}^{t-k}{\vartheta }_2\right)+\sum _{k=1}^{t-1}\varphi \left({{\vartheta }_1}^k\right)-(2t-1)}& \varphi \left({{\vartheta }_1}^t\right)-1\end{array}\right\}\bigcup {\sigma }_{\alpha }\left(Y\right).\end{array}$$

The roots of matrix (7)’s characteristic polynomial can be used to determine the remaining eigenvalues

$$Y=\left[\begin{array}{ccccccc}a& \dots & \delta \sqrt{m_{{\vartheta }_1}{m}_{{{\vartheta }_1}^t}}& \delta \sqrt{m_{{\vartheta }_1}{m}_{{\vartheta }_2}}& \delta \sqrt{m_{{\vartheta }_1}{m}_{{\vartheta }_1{\vartheta }_2}}& \dots & \delta \sqrt{m_{{\vartheta }_1}{m}_{{{\vartheta }_1}^{t-1}{\vartheta }_2}}\\ \delta \sqrt{m_{{\vartheta }_1}{m}_{{{\vartheta }_1}^2}}& \dots & \delta \sqrt{m_{{{\vartheta }_1}^2}{m}_{{{\vartheta }_1}^t}}& \delta \sqrt{m_{{{\vartheta }_1}^2}{m}_{{\vartheta }_2}}& \delta \sqrt{m_{{{\vartheta }_1}^2}{m}_{{\vartheta }_1{\vartheta }_2}}& \dots & \delta \sqrt{m_{{{\vartheta }_1}^2}{m}_{{{\vartheta }_1}^{t-1}{\vartheta }_2}}\\ ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ \delta \sqrt{m_{{\vartheta }_1}{m}_{{{\vartheta }_1}^t}}& \dots & b& \delta \sqrt{m_{{\vartheta }_2}{m}_{{{\vartheta }_1}^t}}& \delta \sqrt{m_{{\vartheta }_1{\vartheta }_2}{m}_{{{\vartheta }_1}^t}}& \dots & \delta \sqrt{m_{{{\vartheta }_1}^{t-1}{\vartheta }_2}{m}_{{{\vartheta }_1}^t}}\\ \delta \sqrt{m_{{\vartheta }_1}{m}_{{\vartheta }_2}}& \dots & \delta \sqrt{m_{{{\vartheta }_1}^t}{m}_{{\vartheta }_2}}& c& \delta \sqrt{m_{{\vartheta }_1{\vartheta }_2}{m}_{{\vartheta }_2}}& \dots & \delta \sqrt{m_{{\vartheta }_2}{m}_{{{\vartheta }_1}^{t-1}{\vartheta }_2}}\\ \delta \sqrt{m_{{\vartheta }_1}{m}_{{\vartheta }_1{\vartheta }_2}}& \dots & \delta \sqrt{m_{{\vartheta }_1{\vartheta }_2}{m}_{{{\vartheta }_1}^t}}& \delta \sqrt{m_{{\vartheta }_2}{m}_{{\vartheta }_1{\vartheta }_2}}& d& \dots & \delta \sqrt{m_{{\vartheta }_1{\vartheta }_2}{m}_{{{\vartheta }_1}^{t-1}{\vartheta }_2}}\\ ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ \delta \sqrt{m_{{\vartheta }_1}{m}_{{{\vartheta }_1}^{t-1}{\vartheta }_2}}& \dots & \delta \sqrt{m_{{{\vartheta }_1}^{t-1}{\vartheta }_2}{m}_{{{\vartheta }_1}^t}}& \delta \sqrt{m_{{{\vartheta }_1}^{t-1}{\vartheta }_2}{m}_{{\vartheta }_2}}& \delta \sqrt{m_{{{\vartheta }_1}^{t-1}{\vartheta }_2}{m}_{{\vartheta }_1{\vartheta }_2}}& \dots & e\end{array}\right]$$

(7)

where $a=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^{t-1}{\vartheta }_2\right)(\alpha -1)-1,b=\alpha \left|V\right|-\varphi \left({\vartheta }_2\right)(\alpha -1)-1,c=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^t\right)(\alpha -2)-2,d=\alpha \left|V\right|-\varphi \left({{\vartheta }_1}^{t-1}\right)(\alpha -1)-1,e=\alpha \left|V\right|-\varphi \left({\vartheta }_1\right)(\alpha -1)-1,\left|V\right|={\displaystyle \sum _{k=1}^t\varphi \left({\vartheta }_1^k\right)+\sum _{k=0}^{t-1}\varphi \left({{\vartheta }_1}^k{\vartheta }_2\right)}$ and $\delta =(1-\alpha ).$ □

When we choose $t=2$ in Theorem 5, the conclusion can be derived.

Corollary 3.

For distinct primes ${\vartheta }_1,{\vartheta }_2$ and $n={{\vartheta }_1}^2{\vartheta }_2$, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ is given by

$$\sigma (D_{\alpha }(W\Gamma ({\mathbb{Z}}_{{{\vartheta }_1}^2{\vartheta }_2})))=\left\{\begin{array}{cc}\alpha \left|V\right|-1& \alpha \left|V\right|+\alpha \varphi \left({{\vartheta }_1}^2\right)-2\\ \varphi \left({\vartheta }_1{\vartheta }_2\right)+\varphi \left({\vartheta }_1\right)+\varphi \left({\vartheta }_2\right)-3& \varphi \left({{\vartheta }_1}^2\right)-1\end{array}\right\}.$$

The cardinality $\left|V\right|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^2{\vartheta }_2}\right)$ is given by $\varphi \left({\vartheta }_1{\vartheta }_2\right)+\varphi \left({\vartheta }_1^2\right)+\varphi \left({\vartheta }_2\right)+\varphi \left({\vartheta }_1\right)$ and the remaining four $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ are the eigenvalues of matrix (8).

Proof. 

Let $n={{\vartheta }_1}^2{\vartheta }_2$, where ${\vartheta }_1<{\vartheta }_2$, note that ${\delta }_{{{\vartheta }_1}^2{\vartheta }_2}^{*}$ is complete graph on vertices $\{{\vartheta }_1,{\vartheta }_2,{{\vartheta }_1}^2,{\vartheta }_1{\vartheta }_2\}.$ By Lemma 4, we have $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^2{\vartheta }_2}\right)$ = ${\delta }_{{{\vartheta }_1}^2{\vartheta }_2}^{*}[W\Gamma \left(A_{{\vartheta }_1}\right),W\Gamma \left(A_{{\vartheta }_2}\right),W\Gamma \left(A_{{{\vartheta }_1}^2}\right),$$W\Gamma \left(A_{{\vartheta }_1{\vartheta }_2}\right)]$. Therefore, by Lemma 2 and Corollary 1, we have $W\Gamma \left(A_{{\vartheta }_1}\right)=K_{\varphi \left({\vartheta }_1{\vartheta }_2\right)},W\Gamma \left(A_{{\vartheta }_2}\right)$$={\overline{K}}_{\varphi \left({{\vartheta }_1}^2\right)},W\Gamma \left(A_{{{\vartheta }_1}^2}\right)=K_{\varphi \left({\vartheta }_2\right)}$ and $W\Gamma \left(A_{{\vartheta }_1{\vartheta }_2}\right)=K_{\varphi \left({\vartheta }_1\right)}$.

The cardinality $\left|V\right|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^2{\vartheta }_2}\right)$ is given by $\varphi \left({\vartheta }_1{\vartheta }_2\right)+\varphi \left({\vartheta }_1^2\right)+\varphi \left({\vartheta }_2\right)+\varphi \left({\vartheta }_1\right)$ and $m_1=\varphi \left({\vartheta }_1{\vartheta }_2\right),m_2=\varphi \left({{\vartheta }_1}^2\right),m_3=\varphi \left({\vartheta }_2\right),m_4=\varphi \left({\vartheta }_1\right)$. It follows that $r_1=\varphi \left({\vartheta }_1{\vartheta }_2\right)-1,r_2=0,r_3=\varphi \left({\vartheta }_2\right)-1$ and $r_4=\varphi \left({\vartheta }_1\right)-1.$ Therefore, by Theorem 1, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{{{\vartheta }_1}^2{\vartheta }_2}\right)$ is

$$\left\{\begin{array}{cc}\alpha \left|V\right|-1& \alpha \left|V\right|+\alpha \varphi \left({{\vartheta }_1}^2\right)-2\\ \varphi \left({\vartheta }_1{\vartheta }_2\right)+\varphi \left({\vartheta }_1\right)+\varphi \left({\vartheta }_2\right)-3& \varphi \left({{\vartheta }_1}^2\right)-1\end{array}\right\}\bigcup {\sigma }_{\alpha }\left(Y\right).$$

The matrix’s characteristic polynomial of the matrix given in (8) can be used to determine the remaining four eigenvalues,

$$Y=\left[\begin{array}{cccc}A\prime & \delta \sqrt{\varphi \left({\vartheta }_1{\vartheta }_2\right)\varphi \left({\vartheta }_1^2\right)}& \delta \sqrt{\varphi \left({\vartheta }_1{\vartheta }_2\right)\varphi \left({\vartheta }_2\right)}& \delta \sqrt{\varphi \left({\vartheta }_1{\vartheta }_2\right)\varphi \left({\vartheta }_1\right)}\\ \delta \sqrt{\varphi \left({\vartheta }_1^2\right)\varphi \left({\vartheta }_1{\vartheta }_2\right)}& B\prime & \delta \sqrt{\varphi \left({\vartheta }_1^2\right)\varphi \left({\vartheta }_2\right)}& \delta \sqrt{\varphi \left({\vartheta }_1^2\right)\varphi \left({\vartheta }_1\right)}\\ \delta \sqrt{\varphi \left({\vartheta }_2\right)\varphi \left({\vartheta }_1{\vartheta }_2\right)}& \delta \sqrt{\varphi \left({\vartheta }_2\right)\varphi \left({\vartheta }_1^2\right)}& C\prime & \delta \sqrt{\varphi \left({\vartheta }_2\right)\varphi \left({\vartheta }_1\right)}\\ \delta \sqrt{\varphi \left({\vartheta }_1\right)\varphi \left({\vartheta }_1{\vartheta }_2\right)}& \delta \sqrt{\varphi \left({\vartheta }_1^2\right)\varphi \left({\vartheta }_1\right)}& \delta \sqrt{\varphi \left({\vartheta }_1\right)\varphi \left({\vartheta }_2\right)}& D\prime \end{array}\right]$$

(8)

where $\delta =(1-\alpha ),A\prime =\alpha \left|V\right|-\varphi \left({\vartheta }_1{\vartheta }_2\right)\left(\alpha -1\right)-1,B\prime =\alpha \left|V\right|-\varphi \left({\vartheta }_1^2\right)\left(\alpha -2\right)-2,C\prime =\alpha \left|V\right|-\varphi \left({\vartheta }_2\right)\left(\alpha -1\right)-1$ and $D\prime =\alpha \left|V\right|-\varphi \left({\vartheta }_1\right)\left(\alpha -1\right)-1.$ □

Example 1.

The $D_{\alpha }$ spectrum of the weakly zero-divisor graph of ${\mathbb{Z}}_{20}$ is

$$\sigma \left(D_{\alpha }(W\Gamma \left({\mathbb{Z}}_{20}\right))\right)=\left\{\begin{array}{cc}11\alpha -1& 13\alpha -2\\ 6& 1\end{array}\right\}.$$

The remaining four $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{20}\right)$ (Figure 1) are the roots of the characteristic polynomial of matrix (9).

Figure 1. (a) Weakly zero-divisor graph $W\Gamma \left({\mathbb{Z}}_{20}\right)$. (b) Proper divisor graph ${\delta }_{20}^{*}$.

From Corollary 3, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{20}\right)$ is given by

$${\sigma }_{\alpha }(W\Gamma \left({\mathbb{Z}}_{20}\right))=\left\{\begin{array}{cc}11\alpha -1& 13\alpha -2\\ 6& 1\end{array}\right\}\bigcup {\sigma }_{\alpha }\left(Y\right).$$

And the roots of characteristic polynomial of the matrix (9) provided below, are remaining four $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{20}\right)$

$$Y=\left[\begin{array}{cccc}3+7\alpha & 2\delta \sqrt{2}& 4& 2\delta \\ 2\delta \sqrt{2}& 9\alpha +2& 2\delta \sqrt{2}& \delta \sqrt{2}\\ 4& 2\delta \sqrt{2}& 3+7\alpha & 2\delta \\ 2\delta & \delta \sqrt{2}& 2\delta & 10\alpha \end{array}\right]$$

(9)

where $\delta =(1-\alpha ).$

Theorem 6.

Let $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes. Then, the $D_{\alpha }$ spectrum of the $W\Gamma \left({\mathbb{Z}}_n\right)$ consists of eigenvalues

$$\{\begin{array}{cccc}\alpha |V|-1& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_1}}\right)\right)+\alpha |V|-2& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_2}}\right)\right)+\alpha |V|-2& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_3}}\right)\right)+\alpha |V|-2\\ {\displaystyle \sum _{c_i\ne {\vartheta }_i}\varphi \left({\displaystyle \frac{n}{c_i}}\right)}-(\tau \left(n\right)-2-t)& \varphi \left({\displaystyle \frac{n}{{\vartheta }_1}}\right)-1& \varphi \left({\displaystyle \frac{n}{{\vartheta }_2}}\right)-1& \varphi \left({\displaystyle \frac{n}{{\vartheta }_3}}\right)-1\end{array}$$

$$\begin{array}{ccc}\dots & \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_{t-1}}}\right)\right)+\alpha |V|-2& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_t}}\right)\right)+\alpha |V|-2\\ \dots & \varphi \left({\displaystyle \frac{n}{{\vartheta }_{t-1}}}\right)-1& \varphi \left({\displaystyle \frac{n}{{\vartheta }_t}}\right)-1\end{array}\}.$$

The cardinality $|V|$ of the vertex set V of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ is given by $|V|={\displaystyle \sum _{i=1}^{\tau \left(n\right)-2}\varphi \left({\displaystyle \frac{n}{c_i}}\right)}$ and the characteristic polynomial of matrix (10) provides the remaining eigenvalues. Also, if the eigenvalues of matrix (10) are integers, then the weakly zero divisor graph of ${\mathbb{Z}}_n$ for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$ is $D_{\alpha }$ integral.

Proof. 

Suppose that $n={\vartheta }_1{\vartheta }_2\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$, where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes. Let $\beta =\{{\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_t\}.$ Then, by Lemma 3, the following conclusions can be drawn: for each $c_i\in \beta$, we have $W\Gamma \left(A_{c_i}\right)={\overline{K}}_{\varphi \left({\displaystyle \frac{n}{c_i}}\right)}$, and for $c_j\notin \beta$, we have $W\Gamma \left(A_{c_i}\right)=K_{\varphi \left({\displaystyle \frac{n}{c_i}}\right)}.$ The cardinality $|V|$ of the vertex set V of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ is ${\displaystyle |V|=\sum _{i=1}^{\tau \left(n\right)-2}\varphi \left({\displaystyle \frac{n}{c_i}}\right)}.$ Also note that for $1\le i,j\le \tau \left(n\right)-2$, we have $m_{c_i}=\varphi \left({\displaystyle \frac{n}{c_i}}\right),m_{c_j}=\varphi \left({\displaystyle \frac{n}{c_j}}\right)$ for all $c_i\in \beta$ and $c_j\notin \beta .$ Therefore, by Theorem 1, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_n\right)$ is

$$\{\begin{array}{cccc}\alpha |V|-1& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_1}}\right)\right)+\alpha |V|-2& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_2}}\right)\right)+\alpha |V|-2& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_3}}\right)\right)+\alpha |V|-2\\ {\displaystyle \sum _{c_i\ne {\vartheta }_i}\varphi \left({\displaystyle \frac{n}{c_i}}\right)-(\tau \left(n\right)-2-t)}& \varphi \left({\displaystyle \frac{n}{{\vartheta }_1}}\right)-1& \varphi \left({\displaystyle \frac{n}{{\vartheta }_2}}\right)-1& \varphi \left({\displaystyle \frac{n}{{\vartheta }_3}}\right)-1\end{array}$$

$$\begin{array}{cccc}& \dots & \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_{t-1}}}\right)\right)+\alpha |V|-2& \alpha \left(\varphi \left({\displaystyle \frac{n}{{\vartheta }_t}}\right)\right)+\alpha |V|-2\\ & \dots & \varphi \left({\displaystyle \frac{n}{{\vartheta }_{t-1}}}\right)-1& \varphi \left({\displaystyle \frac{n}{{\vartheta }_t}}\right)-1\end{array}\}.$$

Note that all these eigenvalues are integers. And the roots of matrix (10)’s characteristic polynomial can be used to determine the remaining eigenvalues

$$\left[\begin{array}{cccccc}{\mu }_{c_1}\left(\alpha \right)& \dots & \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_1}}\right)\varphi \left({\displaystyle \frac{n}{c_t}}\right)}& \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_1}}\right)\varphi \left({\displaystyle \frac{n}{c_{t+1}}}\right)}& \dots & \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_1}}\right)\varphi \left({\displaystyle \frac{n}{c_{\tau \left(n\right)-2}}}\right)}\\ ⋮& \ddots & ⋮& ⋮& ⋮& ⋮\\ \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_1}}\right)\varphi \left({\displaystyle \frac{n}{c_t}}\right)}& \dots & {\mu }_{c_t}\left(\alpha \right)& \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_t}}\right)\varphi \left({\displaystyle \frac{n}{c_{t+1}}}\right)}& \dots & \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_t}}\right)\varphi \left({\displaystyle \frac{n}{c_{\tau \left(n\right)-2}}}\right)}\\ \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_1}}\right)\varphi \left({\displaystyle \frac{n}{c_{t+1}}}\right)}& \dots & \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_t}}\right)\varphi \left({\displaystyle \frac{n}{c_{t+1}}}\right)}& {\mu }_{c_{t+1}}\left(\alpha \right)& \dots & \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_{t+1}}}\right)\varphi \left({\displaystyle \frac{n}{c_{\tau \left(n\right)-2}}}\right)}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_1}}\right)\varphi \left({\displaystyle \frac{n}{c_{\tau \left(n\right)-2}}}\right)}& \dots & \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_t}}\right)\varphi \left({\displaystyle \frac{n}{c_{\tau \left(n\right)-2}}}\right)}& \delta \sqrt{\varphi \left({\displaystyle \frac{n}{c_{t+1}}}\right)\varphi \left({\displaystyle \frac{n}{c_{\tau \left(n\right)-2}}}\right)}& \dots & {\mu }_{c_{\tau \left(n\right)-2}}\left(\alpha \right)\end{array}\right]$$

(10)

where $\delta =(1-\alpha )$ and

$${\mu }_{c_i}\left(\alpha \right)=\left\{\begin{array}{cc}\alpha |V|-\varphi \left({\displaystyle \frac{n}{c_i}}\right)(\alpha -2)-2,\hfill & c_i\in \beta \hfill \\ \alpha |V|-\varphi \left({\displaystyle \frac{n}{c_i}}\right)(\alpha -1)-1,\hfill & c_i\notin \beta .\hfill \end{array}\right.$$

If the eigenvalues of matrix (10) are integers, then the weakly zero divisor graph of ${\mathbb{Z}}_n$ for $n={\vartheta }_1{\vartheta }_2{\vartheta }_3\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$ is $D_{\alpha }$ integral. □

Example 2.

The $D_{\alpha }$ spectrum of the weakly zero-divisor graph of ${\mathbb{Z}}_{42}$, shown in Figure 2, is

$$\sigma \left(D_{\alpha }(W\Gamma \left({\mathbb{Z}}_{42}\right))\right)=\left\{\begin{array}{cccc}29\alpha -1& 41\alpha -2& 35\alpha -2& 31\alpha -1\\ 6& 11& 5& 1\end{array}\right\}.$$

The six remaining $D_{\alpha }$ eigenvalues of the graph $W\Gamma \left({\mathbb{Z}}_{42}\right)$ are the eigenvalues of matrix (11).

The proper divisors of 42 are 2, 3, 7, 6, 14, and $21.$ Note that ${\delta }_{42}^{*}$ is a complete graph on vertices $2,3,7,6,14$, and 21. Now, by Lemma 4, we have $W\Gamma \left(Z_{42}\right)={\delta }_{42}^{*}[W\Gamma \left(A_2\right),W\Gamma \left(A_3\right),W\Gamma \left(A_7\right),W\Gamma \left(A_6\right),W\Gamma \left(A_{14}\right),W\Gamma \left(A_{21}\right)]$. Therefore, by Lemma 2 and Corollary 1, we have $W\Gamma \left(A_2\right)=\overline{K_{12}},W\Gamma \left(A_3\right)={\overline{K}}_6,W\Gamma \left(A_7\right)={\overline{K}}_2,W\Gamma \left(A_6\right)=K_6,W\Gamma \left(A_{14}\right)=K_2$ and $W\Gamma \left(A_{21}\right)=K_1$. The cardinality $|V|$ of the vertex set V of $W\Gamma \left({\mathbb{Z}}_{42}\right)$ is 19. Now, according to the proper divisor sequence, we have $m_1=12,m_2=6,m_3=2,m_4=6,m_5=2,m_6=1$, and further, we have $r_1=r_2=r_3=0,r_4=5,r_5=1$, and $r_6=0.$ Consequently, the $D_{\alpha }$ spectrum of the graph $W\Gamma \left({\mathbb{Z}}_{42}\right)$ is given by Theorem 1.

$$\sigma \left(D_{\alpha }(W\Gamma \left({\mathbb{Z}}_{42}\right))\right)=\left\{\begin{array}{cccc}29\alpha -1& 41\alpha -2& 35\alpha -2& 31\alpha -1\\ 6& 11& 5& 1\end{array}\right\}.$$

And the matrix’s characteristic polynomial can be used to determine the remaining six eigenvalues

$$Y=\left[\begin{array}{cccccc}17\alpha +22& 6\delta \sqrt{2}& 2\delta \sqrt{6}& 6\delta \sqrt{2}& 2\delta \sqrt{6}& 2\delta \sqrt{3}\\ 6\delta \sqrt{2}& 23\alpha +10& 2\delta \sqrt{3}& 6\delta & 2\delta \sqrt{3}& \delta \sqrt{6}\\ 2\delta \sqrt{6}& 2\delta \sqrt{3}& 27\alpha +2& 2\delta \sqrt{3}& 2\delta & \delta \sqrt{2}\\ 6\delta \sqrt{2}& 6\delta & 2\delta \sqrt{3}& 23\alpha +5& 2\delta \sqrt{3}& \delta \sqrt{6}\\ 2\delta \sqrt{6}& 2\delta \sqrt{3}& 2\delta & 2\delta \sqrt{3}& 1+27\alpha & \delta \sqrt{2}\\ 2\delta \sqrt{3}& \delta \sqrt{6}& \delta \sqrt{2}& \delta \sqrt{6}& \delta \sqrt{2}& 28\alpha \end{array}\right]$$

(11)

where $\delta =(1-\alpha ).$

Figure 2. Weakly zero-divisor graph $W\Gamma \left({\mathbb{Z}}_{42}\right)$.

5. Conclusions and Further Work

In this study, we have explored the $D_{\alpha }$ spectrum of the weakly zero-divisor graph $W\Gamma \left({\mathbb{Z}}_n\right))$ for a general class of n, where $n={\vartheta }_1{\vartheta }_2\dots {\vartheta }_t{\eta }_1^{d_1}{\eta }_2^{d_2}\dots {\eta }_s^{d_s}(d_i\ge 2,t\ge 1,s\ge 0)$ and where ${\vartheta }_i$’s and ${\eta }_i$’s are the distinct primes. For this, we use the concept of new results on the $D_{\alpha }$-matrix of connected graphs, which was introduced by Diaz et al. [4]. We obtain the $D_{\alpha }$ eigenvalues for several arrangements by using thorough calculations and the basic algebraic properties of the weakly zero-divisor graph. This study shows how the algebraic structure of ${\mathbb{Z}}_n$ and the spectral features of its associated graph interact, building on earlier findings regarding particular classes of n.

The results show that the $D_{\alpha }$ spectrum contains important information regarding the basic structure of weakly zero divisors. Specifically, the eigenvalue distributions and characteristic polynomials provide clarity on the modular arithmetic and divisors that underlie ${\mathbb{Z}}_n$. The findings further support the significance of spectral graph theory in algebraic contexts by validating the existence of distinctive spectral patterns in specific classes of weakly zero-divisor graphs.

This study suggests various exciting paths for further exploration. One possible direction is to broaden the spectral analysis to encompass wider categories of finite commutative rings, with the goal of uncovering more profound connections between their algebraic characteristics and spectral parameters. An additional area worth exploring involves the investigation of further graph invariants, including the spectral radius, chromatic number, and connectivity, along with their relationships to the $D_{\alpha }$ spectrum of weakly zero-divisor graphs. The spectra for higher powers of primes and rings with multiple prime factors could also be analyzed using sophisticated computer approaches, which could reveal complex patterns and features. Furthermore, it is necessary to conduct through research since the spectrum characteristics of weakly zero-divisor graphs may have useful implications in coding theory, cryptography, and error detection systems. It may be possible to identify significant similarities and differences between weakly zero-divisor graphs and other algebraically defined graphs, such as unit graphs or co-maximal graphs, by comparing their spectral properties.

Further one can calculate $D_{\alpha }$ spectrum for co-zero divisor graphs, unit graphs, co-maximal graphs, and many more such graphs. This research not only deepens the theoretical insights into weakly zero-divisor graphs but also creates a strong basis for interdisciplinary studies that combine algebra, graph theory, and computational techniques.