On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring ${\mathbb{Z}}_{{\mathit{p}}_{\mathbf{1}}^{{\mathit{T}}_{\mathbf{1}}}{\mathit{p}}_{\mathbf{2}}^{{\mathit{T}}_{\mathbf{2}}}}$

Abstract

The zero-divisor graph of a commutative ring $\mathfrak{R}$ with a nonzero identity, denoted by $\Gamma \left(\mathfrak{R}\right)$, is an undirected graph where the vertex set $Z{\left(\mathfrak{R}\right)}^{*}$ consists of all nonzero zero-divisors of $\mathfrak{R}$. Two distinct vertices a and b in $\Gamma \left(\mathfrak{R}\right)$ are adjacent if and only if $ab=0$. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for specific values of $\mathfrak{n}$ in the set $\{pq,p^{2}q,p^3,p^4\}$, where p and q are distinct primes satisfying $p<q$. Motivated by their work, this article investigates the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for a more general class of $\mathfrak{n}$, where $\mathfrak{n}$ is represented as $p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ being distinct primes $(p_1<p_2)$, and $T_1,T_2$ are positive integers.

1. Introduction

Consider a commutative ring denoted as $\mathfrak{R}$, which has a nonzero identity element denoted by $1\ne 0$. An element $a\in \mathfrak{R}$ is called a zero-divisor if there exists another nonzero element b in $\mathfrak{R}$ such that their product $ab=0$. The set of all zero-divisors in $\mathfrak{R}$ is denoted by $Z\left(\mathfrak{R}\right)$, while $Z{\left(\mathfrak{R}\right)}^{*}$ denotes the subset of $Z\left(\mathfrak{R}\right)$ containing all nonzero elements. Zero-divisors play a crucial role in understanding the structural properties of commutative rings, capturing the non-regular elements that are not units.

The symbol ${\mathbb{Z}}_{\mathfrak{n}}$ represents the set of integers modulo $\mathfrak{n}$, where $\mathfrak{n}$ is a positive integer. This ring serves as a fundamental example of a finite commutative ring, with its structure heavily influenced by the factorization of $\mathfrak{n}$ into its prime power components. The elements of ${\mathbb{Z}}_{\mathfrak{n}}$ are denoted by $\{0,1,2,\dots ,\mathfrak{n}-1\}$, with addition and multiplication defined modulo $\mathfrak{n}$. The set $Z\left({\mathbb{Z}}_{\mathfrak{n}}\right)$ contains all elements with nontrivial divisors in ${\mathbb{Z}}_{\mathfrak{n}}$, while the subset $Z{\left({\mathbb{Z}}_{\mathfrak{n}}\right)}^{*}$ focuses on the nonzero elements. The study of $Z{\left({\mathbb{Z}}_{\mathfrak{n}}\right)}^{*}$ plays a significant role in algebraic and graph-theoretical investigations.

In the study of commutative rings, I. Beck [1] introduced the concept of zero-divisor graphs in 1988. Initially, all elements of the ring were considered as vertices of the graph. Later, Anderson and Livingston [2] refined this definition in 1999, focusing solely on the nonzero zero-divisors as graph vertices. The set of vertices in the zero-divisor graph is represented by $Z{\left(\mathfrak{R}\right)}^{*}$. Two distinct elements u and v in $Z{\left(\mathfrak{R}\right)}^{*}$ are adjacent if and only if their product $uv=0$. This graph is denoted by $\Gamma \left(\mathfrak{R}\right)$.

The study of $\Gamma \left(\mathfrak{R}\right)$ has unveiled deep connections between the algebraic properties of the ring $\mathfrak{R}$ and the combinatorial structure of the graph. A key observation is that $\mathfrak{R}$ is an integral domain if and only if $\Gamma \left(\mathfrak{R}\right)$ is an empty graph. This characteristic underscores the utility of $\Gamma \left(\mathfrak{R}\right)$ in distinguishing integral domains from other commutative rings. Moreover, the structure of $\Gamma \left(\mathfrak{R}\right)$ reveals valuable insights into the zero-divisors of $\mathfrak{R}$, including their quantity, distribution, and interaction patterns.

The zero-divisor graph has been instrumental in studying various algebraic structures, including localization, ideal theory, and factorization properties of rings. Moreover, the interplay between graph invariants and ring-theoretic properties has been a focal point of research. Parameters such as diameter, girth, chromatic number, and degree sequences of $\Gamma \left(\mathfrak{R}\right)$ have been analyzed to deduce properties of the underlying ring. These studies have been further extended to specific classes of commutative rings, such as ${\mathbb{Z}}_n$, yielding numerous noteworthy results. For a detailed exploration of zero-divisor graphs and their applications, we direct readers to [2,3] and the related references.

If we use V to represent the set of vertices in the graph $\mathcal{F}$ and E to indicate the set of edges in the same graph $\mathcal{F}$, we can express $\mathcal{F}=(V,E)$. The set of vertices within $\mathcal{F}$ that share an edge with vertex $\mathfrak{a}$ is formally referred to as the neighborhood of $\mathfrak{a}$, and is denoted as $N_{\mathcal{F}}\left(\mathfrak{a}\right)$. The number of edges that are connected to a particular vertex $\mathfrak{a}$ within the vertex set V is denoted by $deg\left(\mathfrak{a}\right)$. This value is referred to as the degree of the vertex $\mathfrak{a}$. A graph $\mathcal{F}$ is said to be r-regular if $deg\left(\mathfrak{a}\right)=r$ for every vertex $\mathfrak{a}\in V$. The spectrum of graph $\mathcal{F}$ with its eigenvalues and multiplicities is represented by the notation $\sigma \left(\mathcal{F}\right)$ for any graph $\mathcal{F}$. If vertices u and v are connected within the graph $\mathcal{F}$, this is denoted as $u\sim v$. The complete graph comprising u vertices is symbolized as ${\mathcal{K}}_u$, while the complete-bipartite graph with a pair of sets of vertices of sizes $(u,v)$ is denoted as ${\mathcal{K}}_{u,v}$. It is important to note that sources [4,5] also include further notations and terms that may not have been previously defined.

For a square matrix A with unique eigenvalues ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_k$ occurring with multiplicities ${\mathfrak{n}}_1,{\mathfrak{n}}_2,\dots ,{\mathfrak{n}}_k$, respectively, then the $\mathit{spectrum}$ of the matrix A, denoted by $\sigma \left(A\right)$ is defined as

$$\sigma \left(A\right)=\left\{\begin{array}{cccc}{\alpha }_1& {\alpha }_2& \cdots & {\alpha }_k\\ {\mathfrak{n}}_1& {\mathfrak{n}}_2& \cdots & {\mathfrak{n}}_k\end{array}\right\}.$$

The square matrix of dimension $\mathfrak{n}$, which is referred to as the adjacency matrix of the graph $\mathcal{F}$, is denoted by $A\left(\mathcal{F}\right)$. This matrix is specified in the following manner:

$$A\left(\mathcal{F}\right)=\left(a_{ij}\right)=\left\{\begin{array}{cc}1,\hfill & v_i{v}_j\in E\left(\mathcal{F}\right),\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

For more comprehensive information regarding adjacency and Laplacian spectra, please refer to sources [6,7,8]. Additionally, the matrix denoted as $\mathcal{L}\left(\mathcal{F}\right)$, representing the graph $\mathcal{F}$, has a normalized Laplacian matrix that is defined in the following manner:

$$\mathcal{L}(\mathfrak{a},\mathfrak{b})=\left\{\begin{array}{cc}1,\hfill & if\mathfrak{a}=\mathfrak{b}anddeg\left(\mathfrak{b}\right)\ne 0,\hfill \\ -{\displaystyle \frac{1}{\sqrt{deg\left(\mathfrak{a}\right)deg\left(\mathfrak{b}\right)}}},\hfill & if\mathfrak{a}and\mathfrak{b}areadjacent,\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

In fact, we have

$$\begin{array}{cc}\hfill \mathcal{L}\left(\mathcal{F}\right)=& Deg{\left(\mathcal{F}\right)}^{-1/2}L\left(\mathcal{F}\right)Deg{\left(\mathcal{F}\right)}^{-1/2}\hfill \\ \hfill =& Deg{\left(\mathcal{F}\right)}^{-1/2}(Deg\left(\mathcal{F}\right)-A\left(\mathcal{F}\right))Deg{\left(\mathcal{F}\right)}^{-1/2}\hfill \\ \hfill =& I-Deg{\left(\mathcal{F}\right)}^{-1/2}A\left(\mathcal{F}\right)Deg{\left(\mathcal{F}\right)}^{-1/2},\hfill \end{array}$$

with the convention $Deg{\left(\mathcal{F}\right)}^{-1}(\mathfrak{a},\mathfrak{b})=0$ for $deg\left(\mathfrak{b}\right)=0$. For more on spectrum of graphs, we refer the reader to [9,10,11]. Nazim et al. [12] investigated the normalized Laplacian spectrum of the weakly zero-divisor graph of the ring ${\mathbb{Z}}_{\mathfrak{n}}$, providing valuable insights into the spectral properties of these graphs. Their studies underscore the intricate connections between the graph-theoretical and algebraic properties of weakly zero-divisor graphs.

The normalized Laplacian spectrum of zero-divisor graphs has garnered significant attention for its ability to uncover intricate connections between algebraic structures and graph-theoretical properties. Pirzada et al. [13] studied the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, where $\mathfrak{n}$ belongs to the set $\{pq,p^{2}q,p^3,p^4\}$, with p and q being distinct primes, such that $p<q$. Their work laid the groundwork for understanding spectral properties of specific zero-divisor graphs and inspired further investigation into generalized cases. Building upon this motivation, the present study delves into the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, where $\mathfrak{n}=p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ being distinct primes $(p_1<p_2)$ and $T_1,T_2$ are positive integers.

This article begins with an introduction that outlines the motivation and objectives, followed by a review of the relevant literature. In Section 2, we establish the fundamental concepts and results necessary for the study, including definitions, lemmas, and theorems related to zero-divisors, commutative rings, and their associated graphs. These preliminaries form the foundational framework for deriving the main results. In Section 3, we detail the approach and techniques used in our analysis, including the computation of the normalized Laplacian spectra of the weakly zero-divisor graphs. In Section 4, we focus on the normalized Laplacian spectrum of the zero-divisor graph $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, where $\mathfrak{n}$ is expressed as $p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ being distinct primes $(p_1<p_2)$ and $T_1,T_2$ are positive integers. Detailed computations are provided, including characteristic polynomials and approximate eigenvalues for specific cases. This work concludes in Section 5, summarizing the main findings and suggesting directions for future research.

The computation of characteristic polynomials and approximate eigenvalues for diverse matrices has been facilitated through the utilization of matrixcalc.org.

2. Preliminaries

We begin by introducing key definitions and relevant prior research, which will provide the basis for drawing our main conclusions.

For any positive integer $\mathfrak{n}$, we denote the number of its positive divisors as $\tau \left(\mathfrak{n}\right)$. The notation $x|\mathfrak{n}$ is employed to signify that the number x is a divisor of $\mathfrak{n}$. The number of positive integers less than or equal to $\mathfrak{n}$ that do not share common factors with $\mathfrak{n}$ is denoted as $\varphi \left(\mathfrak{n}\right)$, and is referred to as Euler’s phi function. When representing a positive integer $\mathfrak{n}$ in terms of its prime factors, it is described as being in a state of prime decomposition, which is denoted as $\mathfrak{n}={{\mathfrak{q}}_1}^{{\mathfrak{m}}_1}{{\mathfrak{q}}_2}^{{\mathfrak{m}}_2}\dots {{\mathfrak{q}}_k}^{{\mathfrak{m}}_k}$, where ${\mathfrak{m}}_1,{\mathfrak{m}}_2,\dots ,{\mathfrak{m}}_k$ represent a positive integers, while ${\mathfrak{q}}_1,{\mathfrak{q}}_2,\dots ,{\mathfrak{q}}_k$ stand for different prime numbers.

Lemma 1

([14]). If $\mathfrak{n}={{\mathfrak{q}}_1}^{{\mathfrak{m}}_1}{{\mathfrak{q}}_2}^{{\mathfrak{m}}_2}\cdots {{\mathfrak{q}}_t}^{{\mathfrak{m}}_s}$ is a prime decomposition of $\mathfrak{n}$, then $\tau \left(\mathfrak{n}\right)=({\mathfrak{m}}_1+1)({\mathfrak{m}}_2+1)\cdots ({\mathfrak{m}}_s+1)$.

Theorem 1

([14]). The phi function ϕ of Euler satisfies the following conditions:

(1) 

$\varphi \left(st\right)=\varphi \left(s\right)\varphi \left(t\right)$, whenever s and t are relatively prime.

(2) 

The sum of $\varphi \left(s\right)=n$, whenever s divides n, i.e., ${\displaystyle \sum _{s|n}}\varphi \left(s\right)=n$.

(3) 

For any prime q, ${\displaystyle \sum _{i=1}^m}\varphi \left(q^i\right)=q^m-1$.

An integer s is defined as a proper divisor of $\mathfrak{n}$ if and only if it satisfies $s|\mathfrak{n}$ and $1<s<\mathfrak{n}$. Let ${\delta }_{\mathfrak{n}}$ represent the simple graph having a set of vertices $\{s_1,s_2,\dots ,s_k\}$ associated with $\mathfrak{n}$, where $s_1,s_2,\dots ,s_k$ are the distinct proper divisors of $\mathfrak{n}$. In the graph ${\delta }_{\mathfrak{n}}$, a connection is established between two vertices if and only if the integer $\mathfrak{n}$ is a divisor of the product $s_i{s}_j$. If $\mathfrak{n}$ is expressed as a prime factorization $\mathfrak{n}={p_1}^{{\mathfrak{n}}_1}{p_2}^{{\mathfrak{n}}_2}\dots {p_r}^{{\mathfrak{n}}_r}$, then the number of vertices in the graph ${\delta }_{\mathfrak{n}}$ is given by

$$|V\left({\delta }_{\mathfrak{n}}\right)|=\prod _{i=1}^r({\mathfrak{n}}_i+1)-2.$$

For every integer r where $1\le r\le k$, we examine the sets of elements defined by

$$A_{s_r}=\{x\in {\mathbb{Z}}_{\mathfrak{n}}:(x,\mathfrak{n})=s_r\},$$

where $(x,\mathfrak{n})$ represents the gcd of x and $\mathfrak{n}$. Furthermore, it is evident that $A_{s_r}\cap A_{s_x}=\varphi$ whenever $r\ne x$. This observation suggests that the collections $A_{s_1},A_{s_2},\dots ,A_{s_k}$ are pairwise disjoint and partitions the vertex set of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ as follows:

$$V\left(\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)\right)=A_{s_1}\cup A_{s_2}\cup \cdots \cup A_{s_k}.$$

As per the description of $A_{s_r}$, a vertex from $A{s}_r$ and a vertex from $A{s}_x$ within the graph $\Gamma \left(\mathbb{Z}\mathfrak{n}\right)$ become connected if and only if the integer $\mathfrak{n}$ evenly divides the product $s_r{s}_x$, where both r and x are selected from the set $1,2,\dots ,k$. The subsequent lemma provides insight into the size of $A_{s_r}$.

Lemma 2

([8]). Let $s_r$ be the divisor of $\mathfrak{n}$. Then, $\left|A_{s_r}\right|=\varphi \left({\displaystyle \frac{\mathfrak{n}}{s_r}}\right)$, $1\le r\le k$.

Lemma 3

([6]). Let $s_r$ be the proper divisor of the positive integer $\mathfrak{n}$. Then, the following hold:

(i) 

The induced subgraph $\Gamma \left(A_{s_r}\right)$ of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ on the vertex set $A_{s_r}$ is either the complete graph ${\mathcal{K}}_{\varphi \left({\displaystyle \frac{\mathfrak{n}}{s_r}}\right)}$ or its complement graph ${\overline{\mathcal{K}}}_{\varphi \left({\displaystyle \frac{\mathfrak{n}}{s_r}}\right)}$, for $r\in \{1,2,\dots ,k\}$. Indeed, if and only if $\mathfrak{n}|s_r^2$, $\Gamma \left(A_{s_r}\right)$ is ${\mathcal{K}}_{\varphi \left({\displaystyle \frac{\mathfrak{n}}{s_r}}\right)}$.

(ii) 

When $r\ne x$, for $r,x\in \{1,2,\dots ,k\}$, a vertex of $A_{s_r}$ is either adjacent to all or none of the vertices of $A_{s_x}$ in $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$.

As mentioned earlier, Lemma 3 illustrates that the subgraphs $\Gamma \left(A_{s_r}\right)$ in $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ fall into two categories: complete graphs or empty graphs. The following lemma confirms that $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ can be described as a mixture of complete graphs and their complements.

Lemma 4

([6]). Let $\Gamma \left(A_{s_r}\right)$ be the induced subgraph of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ on the vertex set $A_{s_r}$ for $1\le r\le k$. Then, $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)={\delta }_{\mathfrak{n}}[\Gamma \left(A_{s_1}\right),\Gamma \left(A_{s_2}\right),\dots ,\Gamma \left(A_{s_k}\right)]$.

In terms of the normalized Laplacian spectrum of component ${\mathcal{F}}_r$, the next result gives the normalized Laplacian spectrum of $\mathcal{F}[{\mathcal{F}}_1,{\mathcal{F}}_2,\dots ,{\mathcal{F}}_{\mathfrak{n}}]$ and the eigenvalues of an auxiliary matrix.

Theorem 2

([15]). Let H be a graph with $V\left(H\right)=\{1,2,\dots ,t\}$, and $K_r$’s be $s_r$-regular graphs of order ${\mathfrak{n}}_i$$(i=1,2,\dots ,t)$. If $\mathcal{F}=H[K_1,K_2,\dots ,K_t]$. Then, normalized Laplacian spectrum of $\mathcal{F}$ can be computed as follows:

$${\sigma }_{\mathcal{N}\mathcal{L}}\left(\mathcal{F}\right)=\left(\bigcup _{r=1}^t\left({\displaystyle \frac{{\mathfrak{X}}_i}{{\mathfrak{X}}_i+s_i}}+{\displaystyle \frac{s_i}{{\mathfrak{X}}_i+s_i}}\left({\sigma }_{\mathcal{N}\mathcal{L}}\left(K_i\right)\setminus \left\{0\right\}\right)\right)\right)\bigcup \sigma \left(Y_{\mathcal{N}\mathcal{L}}\left(H\right)\right),$$

where

$${\mathfrak{X}}_i=\left\{\begin{array}{cc}{\displaystyle \sum _{j\in N_H\left(i\right)}}{\mathfrak{n}}_j,\hfill & N_H\left(i\right)\ne \varphi ,\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$$

and

$$Y_{\mathcal{N}\mathcal{L}}\left(H\right)={\left(y_{ij}\right)}_{\mathfrak{n}\times \mathfrak{n}}=\left\{\begin{array}{cc}{\displaystyle \frac{{\mathfrak{X}}_i}{{\mathfrak{X}}_i+s_i}},\hfill & i=j,\hfill \\ \\ -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_i{\mathfrak{n}}_j}{({\mathfrak{X}}_i+s_i)({\mathfrak{X}}_j+s_j)}}},\hfill & ij\in E\left(H\right),\hfill \\ \\ 0\hfill & otherwise.\hfill \end{array}\right.$$

(1)

The number ${\mathfrak{X}}_i$ and the matrix $Y_{\mathcal{N}\mathcal{L}}\left(H\right)$ are only dependent on the graph H.

3. Methodology

Graph theory continues to be a vibrant field of research, offering a bridge between pure and applied mathematics, as well as discrete structures. The methodology adopted in this study builds upon established concepts in algebra and graph theory, employing rigorous mathematical tools to derive new results. Our strategies focus on leveraging existing literature to extend known results and to explore novel aspects of zero-divisor graphs.

In this work, the use of linear algebra and matrix theory is central to the analysis. Spectral graph theory, in particular, serves as a powerful framework for examining the interplay between graph-theoretical and algebraic properties. The normalized Laplacian spectrum is employed as a key tool to capture the structural characteristics of the zero-divisor graph of the ring ${\mathbb{Z}}_{\mathfrak{n}}$.

The primary objective of this study is to analyze the normalized Laplacian spectra of the zero-divisor graph $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for a general class of $\mathfrak{n}$, expressed as $p_1^{T_1}{p}_2^{T_2}$, where $p_1$ and $p_2$ are distinct primes, and $T_1,T_2$ are positive integers. To achieve this, we utilize advanced spectral graph theory techniques and computational tools to compute eigenvalues and derive the characteristic polynomials associated with these graphs.

Building on the work of Pirzada et al. [13], who investigated the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for specific values of $\mathfrak{n}$, this study extends the spectral analysis to broader classes of $\mathfrak{n}$. Moreover, results from Wu et al. [15] on the H-join operation of graphs provide a conceptual foundation for deriving normalized Laplacian spectra in this context. Through these methodologies, this work aims to enhance the understanding of the relationship between the algebraic structure of ${\mathbb{Z}}_{\mathfrak{n}}$ and the spectral properties of its associated zero-divisor graph.

4. Normalized Laplacian Spectrum of the Zero-Divisor Graph $\mathbf{\Gamma }\left({\mathbb{Z}}_{{\mathit{p}}_{\mathbf{1}}^{\mathit{M}}{\mathit{p}}_{\mathbf{2}}^{\mathit{N}}}\right)$

It is important to note that the complete graph ${\mathcal{K}}_{\mathfrak{m}}$ and its complement ${\overline{\mathcal{K}}}_{\mathfrak{m}}$ with $\mathfrak{m}$ vertices, each with multiplicity, possess well-defined normalized Laplacian spectra.

$${\sigma }_{\mathcal{N}\mathcal{L}}\left({\mathcal{K}}_{\mathfrak{m}}\right)=\left\{\begin{array}{cc}0& {\displaystyle \frac{\mathfrak{m}}{\mathfrak{m}-1}}\\ 1& \mathfrak{m}-1\end{array}\right\}\mathrm{and}{\sigma }_{\mathcal{N}\mathcal{L}}\left({\overline{\mathcal{K}}}_{\mathfrak{m}}\right)=\left\{\begin{array}{c}0\\ \mathfrak{m}\end{array}\right\}.$$

By Lemma 3 $,\Gamma \left(A_{s_r}\right)$ is either the complete graph ${\mathcal{K}}_{\varphi \left({\displaystyle \frac{\mathfrak{n}}{s_r}}\right)}$ or its complement graph ${\overline{\mathcal{K}}}_{\varphi \left({\displaystyle \frac{\mathfrak{n}}{s_r}}\right)}$ for $r\in \{1,2,\dots ,t\}$. Therefore, according to the assertion made in Theorem 2, the overall number of eigenvalues associated with the normalized Laplacian of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ can be represented as $\mathfrak{n}-\varphi \left(\mathfrak{n}\right)-1$.

Using Theorem 2, we can utilize the given diagram below to calculate the normalized Laplacian spectrum.

Example 1.

Normalized Laplacian spectrum of the zero-divisor graph $\Gamma \left({\mathbb{Z}}_{60}\right)$.

The integer $\mathfrak{n}=60$ has a set of proper divisors, namely: 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. Therefore, ${\delta }_{\mathfrak{n}}$ is the graph $G_{10}:2\sim 30\sim 4\sim 15\sim 20\sim 3,5\sim 12\sim 10\sim 6\sim 20\sim 12,10\sim 30\sim 6$ and $30\sim 12\sim 15$. Expanding the sequence of divisors to arrange the vertices in a specific order and applying Lemma 3 and Lemma 4, we have

$$\Gamma \left({\mathbb{Z}}_{60}\right)={\delta }_{60}[{\overline{\mathcal{K}}}_8,{\overline{\mathcal{K}}}_8,{\overline{\mathcal{K}}}_8,{\overline{\mathcal{K}}}_4,{\overline{\mathcal{K}}}_4,{\overline{\mathcal{K}}}_2,{\overline{\mathcal{K}}}_4,{\overline{\mathcal{K}}}_2,{\overline{\mathcal{K}}}_2,{\mathcal{K}}_1].$$

Since $s_i=0$, then ${\displaystyle \frac{{\mathfrak{X}}_i}{{\mathfrak{X}}_i+s_i}}=1$ for $1\le i\le 10$. So, by using Theorem 2 the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{60}\right)$ has eigenvalue 1 with multiplicity $7+7+7+3+3+1+3+1+1=33$, and the set of 10 remaining eigenvalues can be derived from the subsequent matrix provided below.

$$\left[\begin{array}{cccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{2}{\sqrt{13}}}\\ 0& 1& 0& 0& 0& 0& 0& 0& -{\displaystyle \frac{2}{3}}& 0\\ 0& 0& 1& 0& 0& 0& 0& -{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2}{3}}}& 0& -{\displaystyle \frac{2}{\sqrt{39}}}\\ 0& 0& 0& 1& 0& 0& -{\displaystyle \frac{2}{\sqrt{7}}}& 0& 0& 0\\ 0& 0& 0& 0& 1& -{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2}{5}}}& 0& 0& -{\displaystyle \frac{2}{3\sqrt{5}}}& -\sqrt{{\displaystyle \frac{2}{65}}}\\ 0& 0& 0& 0& -{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2}{5}}}& 1& -{\displaystyle \frac{2}{\sqrt{21}}}& 0& 0& -\sqrt{{\displaystyle \frac{2}{117}}}\\ 0& 0& 0& -{\displaystyle \frac{2}{\sqrt{7}}}& 0& -{\displaystyle \frac{2}{\sqrt{21}}}& 1& -{\displaystyle \frac{2}{7}}& -{\displaystyle \frac{2}{3\sqrt{7}}}& -\sqrt{{\displaystyle \frac{2}{91}}}\\ 0& 0& -{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2}{3}}}& 0& 0& 0& -{\displaystyle \frac{2}{7}}& 1& -{\displaystyle \frac{1}{3\sqrt{7}}}& 0\\ 0& -{\displaystyle \frac{2}{3}}& 0& 0& -{\displaystyle \frac{2}{3\sqrt{5}}}& 0& -{\displaystyle \frac{2}{3\sqrt{7}}}& -{\displaystyle \frac{1}{3\sqrt{7}}}& 1& 0\\ -{\displaystyle \frac{2}{\sqrt{13}}}& 0& -{\displaystyle \frac{2}{\sqrt{39}}}& 0& -\sqrt{{\displaystyle \frac{2}{65}}}& -\sqrt{{\displaystyle \frac{2}{117}}}& -\sqrt{{\displaystyle \frac{2}{91}}}& 0& 0& 1\end{array}\right].$$

The above matrix’s approximated eigenvalues are

$$\{2.05554,1.7686,1.64266,1.39338,1.32517,0.732089,0.518514,0.378559,0.323385,-0.137897\}.$$

Now, we determine the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, for $\mathfrak{n}=p_1^{T_1}{p}_2^{T_2}$, where $p_1<p_2$ are primes and $T_1,T_2$ are positive integers.

Theorem 3.

The normalized Laplacian spectrum of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$, where $T_1=2\mathfrak{m}\le 2\mathfrak{n}=T_2$ consists of the eigenvalues

$$\{\begin{array}{ccccc}1& 1& 1& \dots & 1\\ & & & & \\ \varphi \left(p_1^{T_1-i}{p}_2^{T_2}\right)-1& \varphi \left(p_1^{T_1}{p}_2^{T_2-x}\right)-1& \varphi \left(p_1^{T_1-1}{p}_2^{T_2-x}\right)-1& \dots & \varphi \left(p_1^{\mathfrak{m}}{p}_2^{T_2-k}\right)-1\end{array}$$

$$\begin{array}{cccc}{\displaystyle \frac{p_1^{\mathfrak{m}}{p}_2^l-1}{p_1^{\mathfrak{m}}{p}_2^l-2}}& \dots & 1& {\displaystyle \frac{p_1^{T_1}{p}_2^t-1}{p_1^{T_1}{p}_2^t-2}}\\ & & \\ \varphi \left(p_1^{\mathfrak{m}}{p}_2^{T_2-l}\right)-1& \dots & \varphi \left(p_2^{T_2-k}\right)-1& \varphi \left(p_2^{T_2-t}\right)-1\end{array}\},$$

where $i=1,2,\dots ,\mathfrak{m},\dots ,T_1,x=1,2,\dots ,T_2,k=1,2,\dots ,\mathfrak{n}-1,l=\mathfrak{n},\dots ,2\mathfrak{n}$ and $t=\mathfrak{n},\dots ,2\mathfrak{n}-1$. Also, the remaining normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$ are the roots of the characteristic polynomial of the matrix Equation $\left(1\right)$.

Proof. 

If $\mathfrak{n}=p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ representing distinct prime numbers, and satisfying the conditions $T_1=2\mathfrak{m}\le 2\mathfrak{n}=T_2$, the list of proper divisors of $\mathfrak{n}$ are

$$\begin{array}{cc}\hfill & \{p_1,p_1^2,\dots ,p_1^{\mathfrak{m}},\dots ,p_1^{T_1},p_2,p_2^2,\dots ,p_2^{\mathfrak{n}},\dots ,p_2^{T_2},p_1{p}_2,p_1{p}_2^2,\dots ,p_1{p}_2^{\mathfrak{n}},\dots ,p_1{p}_2^{T_2},\dots ,p_1^{\mathfrak{m}}{p}_2,\hfill \\ \hfill & p_1^{\mathfrak{m}}{p}_2^2,\dots ,p_1^{\mathfrak{m}}{p}_2^{\mathfrak{n}-1},p_1^{\mathfrak{m}}{p}_2^{\mathfrak{n}},\dots ,p_1^{\mathfrak{m}}{p}_2^{T_2},\dots ,p_1^{T_1}{p}_2,p_1^{T_1}{p}_2^2,\dots ,p_1^{T_1}{p}_2^{\mathfrak{n}-1},p_1^{T_1}{p}_2^{\mathfrak{n}},\dots ,p_1^{T_1}{p}_2^{T_2-1}\}.\hfill \end{array}$$

By using Lemma 1, the order of ${\delta }_{\mathfrak{n}}$ is $(T_1+1)(T_2+1)-2=T_1{T}_2+T_1+T_2-1$. From the definition of ${\delta }_{\mathfrak{n}}$, the subsequent adjacency relations are as follows:

$$\begin{array}{cc}\hfill p_1^i\sim & p_1^x{p}_2^{T_2},i+x\ge T_1,\mathrm{for}i=1,2,\dots ,T_1,\hfill \\ \hfill p_2^i\sim & p_1^{T_1}{p}_2^x,i+x\ge T_2,\mathrm{for}i=1,2,\dots ,T_2,\hfill \\ \hfill p_1{p}_2^i\sim & p_1^k{p}_2^x,i+x\ge T_2,\mathrm{for}i=1,2,\dots ,T_2\mathrm{and}k\ge 2\mathfrak{m}-1,\hfill \\ \hfill & ⋮\hfill \\ \hfill p_1^{\mathfrak{m}}{p}_2^i\sim & p_1^k{p}_2^x,i+x\ge T_2,\mathrm{for}i=1,2,\dots ,T_2\mathrm{and}k\ge \mathfrak{m},\hfill \\ \hfill & ⋮\hfill \\ \hfill p_1^{T_1}{p}_2^i\sim & p_1^k{p}_2^x,i+x\ge T_2,\mathrm{for}i=1,2,\dots ,T_2-1\mathrm{and}k\ge 0.\hfill \end{array}$$

In view of Lemma 2, the sizes of sets $A_{s_i}$ for $i=1,2,\dots ,T_1$, $x=1,2,\dots ,T_2$ and $k=1,2,\dots ,T_2-1$ can be determined as follows:

$$\begin{array}{ccccccc}\hfill |A_{p_1^i}|& =\hfill & \hfill \varphi \left(p_1^{T_1-i}{p}_2^{T_2}\right),\left|A_{p_2^x}\right|& =\hfill & \hfill \varphi \left(p_1^{T_1}{p}_2^{T_2-x}\right),\left|A_{p_1{p}_2^x}\right|& =\hfill & \hfill \varphi \left(p_1^{T_1-1}{p}_2^{T_2-x}\right),\\ \hfill \dots ,\left|A_{p_1^{\mathfrak{m}}{p}_2^x}\right|& =\hfill & \hfill \varphi \left(p_1^{\mathfrak{m}}{p}_2^{T_2-x}\right),\dots ,\left|A_{p_1^{T_1-1}{p}_2^x}\right|& =\hfill & \hfill \varphi \left(p_1{p}_2^{T_2-x}\right),\left|A_{p_1^{T_1}{p}_2^k}\right|& =\hfill & \hfill \varphi \left(p_2^{T_2-k}\right).\end{array}$$

As indicated by Lemma 3, the induced subgraphs $\Gamma (A_{s_{p_1^i}})$ are

$$K_i=\left\{\begin{array}{c}\Gamma \left(A_{s_{p_1^i}}\right)={\overline{\mathcal{K}}}_{\varphi \left(p_1^{T_1-i}{p}_2^{T_2}\right)},1\le i\le T_1,\hfill \\ \\ \Gamma \left(A_{s_{p_2^x}}\right)={\overline{\mathcal{K}}}_{\varphi \left(p_1^{T_1}{p}_2^{T_2-x}\right)},1\le x\le T_2,\hfill \\ \\ \Gamma \left(A_{s_{p_1^i{p}_2^x}}\right)={\overline{\mathcal{K}}}_{\varphi \left(p_1^{T_1-i}{p}_2^{T_2-x}\right)},1\le i\le \mathfrak{m}-1\mathrm{and}1\le x\le T_2\hfill \\ \mathrm{or}\mathfrak{m}\le i\le T_1\mathrm{and}1\le x\le \mathfrak{n}-1,\hfill \\ \Gamma \left(A_{s_{p_1^i{p}_2^x}}\right)={\mathcal{K}}_{\varphi \left(p_1^{T_1-i}{p}_2^{T_2-x}\right)},\mathfrak{m}\le i\le T_1\mathrm{and}n\le x\le T_2.\hfill \end{array}\right.$$

(2)

Applications of Theorem 2 yield the values of ${\mathfrak{X}}_i$,

$$\begin{array}{cc}\hfill {\mathfrak{X}}_1=& \varphi \left(p_1\right)=p_1-1,\hfill \\ \hfill {\mathfrak{X}}_2=& \varphi \left(p_1\right)+\varphi \left(p_1^2\right)=p^2-1,\hfill \\ \hfill ⋮& \\ \hfill {\mathfrak{X}}_{\mathfrak{m}}=& \varphi \left(p_1^{\mathfrak{m}}\right)+\varphi \left(p_1^{\mathfrak{m}-1}\right)+\cdots +\varphi \left(p_1\right)=p_1^{\mathfrak{m}}-1,\hfill \\ \hfill ⋮& \\ \hfill {\mathfrak{X}}_{T_1}=& \varphi \left(p_1^{T_1}\right)+\varphi \left(p_1^{T_1-1}\right)+\cdots +\varphi \left(p_1\right)=p_1^{T_1}-1,\hfill \end{array}$$

so we can say that

$${\mathfrak{X}}_i=p_1^i-1,\mathrm{for}i=1,2,\dots ,T_1.$$

For $i\ge \mathfrak{n}$ and $x\ge \mathfrak{n}$, we observe that $\Gamma \left(A_{p_1^i{p}_2^x}\right)$ is adjacent to itself as a vertex of ${\delta }_{\mathfrak{n}}$. Therefore, by adding and subtracting the cardinalities of $\Gamma \left(A_{p_1^i{p}_2^x}\right)$, we obtain ${\mathfrak{X}}_i$. The remaining ${\mathfrak{X}}_i$, computed as described above, are given by

$$\begin{array}{cc}\hfill {\mathfrak{X}}_i=& p_2^x-1,\mathrm{for}i=T_1+1,\dots ,T_1+T_2\mathrm{and}x=1,2,\dots ,\mathfrak{n},\dots ,T_1,\hfill \\ \hfill {\mathfrak{X}}_i=p_1& p_2^x-1,\mathrm{for}i=T_1+T_2+1,\dots ,T_1+T_2\mathrm{and}x=1,2,\dots ,\mathfrak{n},\dots ,T_2,\hfill \\ \hfill & ⋮\hfill \\ \hfill {\mathfrak{X}}_i={p_1}^{\mathfrak{m}}& p_2^x-1,\mathrm{for}i=T_1+\mathfrak{m}{T}_2+1,\dots ,T_1+\mathfrak{m}{T}_2+\mathfrak{n}-1\mathrm{and}x=1,2,\dots ,n-1,\hfill \\ \hfill {\mathfrak{X}}_i={p_1}^{\mathfrak{m}}& p_2^x-1-\varphi \left(p_1^{\mathfrak{m}}{p}_2^x\right),\mathrm{for}i=T_1+\mathfrak{m}{T}_2,\dots ,T_1+(\mathfrak{m}+1)T_2\mathrm{and}x=\mathfrak{n},\dots ,T_2,\hfill \\ \hfill ⋮& \\ \hfill {\mathfrak{X}}_i={p_1}^{T_1}& p_2^x-1,\mathrm{for}i=T_1+T_1{T}_2+1,\dots ,T_1+T_1{T}_2+\mathfrak{n}-1\mathrm{and}x=1,2,\dots ,\mathfrak{n}-1,\hfill \\ \hfill {\mathfrak{X}}_i={p_1}_1^T& p_2^x-1-\varphi \left(p_2^{T_1-x}\right),\mathrm{for}i=T_1+T_1{T}_2+\mathfrak{n},\dots ,T_1+T_1{T}_2+T_2-1\hfill \\ & \mathrm{and}x=\mathfrak{n},\dots ,T_2-1.\hfill \end{array}$$

Now, by using Equation (2) and Theorem 2, we have

$${\displaystyle \frac{{\mathfrak{X}}_i}{s_i+{\mathfrak{X}}_i}}+{\displaystyle \frac{s_i}{s_i+{\mathfrak{X}}_i}}{\alpha }_{ik}\left(K_i\right)={\displaystyle \frac{{\mathfrak{X}}_i}{s_i+{\mathfrak{X}}_i}}+{\displaystyle \frac{s_i}{s_i+{\mathfrak{X}}_i}}{\alpha }_{ik}({\overline{\mathcal{K}}}_{\varphi \left(p_1^{T_1-i}{p}_2^{T_2}\right)})=1,\mathrm{for}i=1,2,\dots ,T_1.$$

Thus, the normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$ is 1 with multiplicity $\varphi \left(p_1^{T_1-i}{p}_2^{T_2}\right)-1$. This is true for $i=1,2,\dots ,T_1$. Using analogous procedures, we find that

$$\left\{\begin{array}{cccc}1& 1& \dots & 1\\ & & & \\ \varphi \left(p_1^{T_1}{p}_2^{T_2-x}\right)-1& \varphi \left(p_1^{T_1-1}{p}_2^{T_2-x}\right)-1& \dots & \varphi \left(p_1^{\mathfrak{m}}{p}_2^{T_2-k}\right)-1\end{array}\right\},$$

are also the normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$. From Equation (2), it follows that $K_i=K_{\varphi \left(p_1^{T_1-i}{p}_2^{T_2-x}\right)}$, where $\mathfrak{m}\le i\le T_1$ and $\mathfrak{n}\le x\le T_2$. Therefore, we obtain

$${\displaystyle \frac{{\mathfrak{X}}_i}{s_i+{\mathfrak{X}}_i}}+{\displaystyle \frac{s_i}{s_i+{\mathfrak{X}}_i}}{\alpha }_{ik}\left(K_i\right)={\displaystyle \frac{p_1^{\mathfrak{m}}{q}^l-1-\varphi \left(p_1^{\mathfrak{m}}{p}_2^l\right)}{p_1^{\mathfrak{m}}{p}_2^l-2}}+{\displaystyle \frac{\varphi \left(p_1^{\mathfrak{m}}{p}_2^l\right)}{p_1^{\mathfrak{m}}{p}_2^l-2}}={\displaystyle \frac{p_1^{\mathfrak{m}}{p}_2^l-1}{p_1^{\mathfrak{m}}{p}_2^l-2}}.$$

It follows that ${\displaystyle \frac{p_1^{\mathfrak{m}}{p}_2^l-1}{p_1^{\mathfrak{m}}{p}_2^l-2}}$ is the normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$ with multiplicity $\varphi \left(p_1^{\mathfrak{m}}{p}_2^l\right)-1$, where $l=\mathfrak{n},\dots ,T_2$. Similarly, we observe that 1 with multiplicity $\varphi \left(p_2^{T_2-k}\right)-1$ and ${\displaystyle \frac{p_1^{T_1}{p}_2^t-1}{p_1^{T_1}{p}_2^t-2}}$ with multiplicity $\varphi (p_2^{T_2-t})-1$ are the normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$ for $k=1,2,\dots ,\mathfrak{n}-1$ and $t=\mathfrak{n},\dots ,T_2-1$. Also, the remaining normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$ are the roots of the characteristic polynomial of the matrix Equation (1). □

An immediate consequence of our first main result is the following corollary.

Corollary 1.

Suppose $\mathfrak{n}=p_1^{2t}$, where $p_1>2$ is prime and $t\ge 2$ is a positive integer. Then, the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ consists of eigenvalue 1 with multiplicity $p_1^{2t-1}-p_1^t-t+1$, the eigenvalue ${\displaystyle \frac{p_1^i-1}{p_1^i-2}}$ with multiplicity $\varphi \left(p_1^{2t-i}\right)-1$, $i=t,t+1,\dots ,2t-2,2t-1$. The other normalized Laplacian eigenvalue of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ are the eigenvalues of matrix Equation (3).

Proof. 

Let $\mathfrak{n}=p_1^{2t}$, for $t\ge 2$ is a positive integer. Then,

$$\{p_1,p_1^2,\dots ,p_1^{t-1},p_1^t,\dots ,p_1^{2t-2},p_1^{2t-1}\}$$

are the proper divisors of $\mathfrak{n}$. We observe that the vertex $p_1$ of ${\delta }_{p_1^{2t}}$ is adjacent to the vertex $p_1^{2t-1}$ and the vertex $p_1^2$ is adjacent to both $p_1^{2t-1}$ and $p_1^{2t-2}$. Thus, in general, for each $j\ge 2t-i$ with $1\le i\le 2t-1$ and $i\ne j$, the vertex $p_1^i$ is adjacent to the vertex $p_1^j$ in ${\delta }_{p_1^{2t}}$. Now, ${\mathfrak{X}}_1=\varphi \left(p_1\right)=p_1-1,{\mathfrak{X}}_2=\varphi \left(p_1\right)+\varphi \left(p_1^2\right)=p_1-1+p_1^2-p_1=p_1^2-1$. So, by using the similar steps and using Theorem 1 (3), we have

$${\mathfrak{X}}_i=p_1^i-1,fori=1,2,\dots ,t-2,t-1.$$

So,

$$({\mathfrak{X}}_1,{\mathfrak{X}}_2,\dots ,{\mathfrak{X}}_{t-2},{\mathfrak{X}}_{t-1})=(p_1-1,p_1^2-1,\dots ,p_1^{t-2}-1,p_1^{t-1}-1).$$

Similarly, for $i=t,t+1,\dots ,2t-2,2t-1$, we obtain

$${\mathfrak{X}}_i=\sum _{j=1}^i\varphi \left(p_1^j\right)-\varphi \left(p_1^{2t-i}\right)=p_1^i-1-\varphi \left(p_1^{2t-i}\right).$$

So,

$$\begin{array}{cc}\hfill ({\mathfrak{X}}_t,{\mathfrak{X}}_{t+1},\dots ,{\mathfrak{X}}_{2t-2},{\mathfrak{X}}_{2t-1})& =(p_1^{t-1}-1,p_1^{t+1}-1-p_1^{t-1}+p_1^{t-1},\hfill \\ \hfill & \dots ,p_1^{2t-2}-1-p_1^2+p,p_1^{2t-1}-p_1).\hfill \end{array}$$

$G_i={\overline{\mathcal{K}}}_{\varphi \left(p_1^{2t-i}\right)}$, for $i=1,2,\dots ,t-2,t-1$ because $\mathfrak{n}$ does not divide ${\left(p_1^i\right)}^2$, for $i=1,2,\dots ,t-2,t-1$ and $G_i={\mathcal{K}}_{\varphi \left(p_1^{2t-i}\right)}$ for $i=t,t+1,\dots ,2t-2,2t-1$ because $\mathfrak{n}$ divides ${\left(p_1^i\right)}^2$, for $i=t,t+1,\dots ,2t-2,2t-1$. This implies that $s_i=0$, for $i=1,2,\dots ,t-2,t-1$, and $s_i=\varphi (p_1^{2t-i}-1)$, for $i=t,t+1,\dots ,2t-2,2t-1$. Further, ${\displaystyle \frac{{\mathfrak{X}}_i}{{\mathfrak{X}}_i+s_i}}=1$, for $i=1,2,\dots ,t-2,t-1$, and ${\displaystyle \frac{{\mathfrak{X}}_i}{{\mathfrak{X}}_i+s_i}}={\displaystyle \frac{p_1^i-1-\varphi \left(p_1^{2t-i}\right)}{p_1^i-2}}$, for $i=t,t+1,\dots ,2t-2,2t-1$. Therefore, by using Theorem 2, we observe that 1 is the normalized Laplacian eigenvalue of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ with multiplicity $\varphi \left(p_1^{2t-1}\right)-1+\varphi \left(p_1^{2t-2}\right)-1+\cdots +\varphi \left(p_1^{t+2}\right)-1+\varphi \left(p_1^{t+1}\right)-1=p_1^{2t-1}-1-p_1^t-t+1$. Also, we have ${\mathfrak{X}}_i+s_i=\varphi \left(p_1^{2t-i}\right)-1+p_1^i-1-\varphi \left(p_1^{2t-i}\right)=p_1^i-2$ for $i=t,t+1,\dots ,2t-2,2t-1$. Therefore, by Theorem 2, ${\displaystyle \frac{p_1^i-1}{p_1^i-2}}$ is the normalized Laplacian eigenvalue of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ with multiplicity $\varphi \left(p_1^{2t-i}\right)-1$. The matrix below provides the other normalized Laplacian eigenvalues of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$

$$\left[\begin{array}{cccccc}{X}_{t-1}& & & Y_{\left(t-1\right)\times \left(t-1\right)}& & \\ & y_t& x_{t,t+1}& \dots & x_{t,2t-2}& x_{t,2t-1}\\ & x_{t+1,t}& y_{t+1}& \dots & x_{t+1,2t-2}& x_{t+1,2t-1}\\ Y^T& ⋮& ⋮& \ddots & ⋮& ⋮\\ & x_{2t-2,t}& x_{2t-2,t+1}& \dots & y_{2t-2}& x_{2t-2,2t-1}\\ & x_{2t-1,t}& x_{2t-1,t+1}& \dots & x_{2t-1,2t-2}& y_{2t-1}\end{array}\right],$$

(3)

where $X_{t-1}=diag(1,1,\dots ,1,1)$,

$$Y=\left[\begin{array}{ccccc}0& 0& \dots & 0& x_{1,2t-1}\\ 0& 0& \dots & x_{2,2t-2}& x_{2,2t-1}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \dots & x_{t-2,2t-2}& x_{t-2,2t-1}\\ 0& x_{t-1,t+1}& \dots & x_{t-1,2t-2}& x_{t,2t-1}\end{array}\right]$$

and $x_{ij}=x_{ji}=-\sqrt{{\displaystyle \frac{{\mathfrak{n}}_i{\mathfrak{n}}_j}{({\mathfrak{X}}_i+s_i)({\mathfrak{X}}_j+s_j)}}}$, for $1\le i,j\le 2t-1$; $y_{ij}={\displaystyle \frac{s_i}{{\mathfrak{X}}_j+s_j}}$, for $i=t,t+1,\dots ,2t-2,2t-1$. □

If $\mathfrak{m}=1$ and $p_2=1$ in Theorem 3, the resulting outcome is that $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ becomes equivalent to the complete graph $K_{\varphi \left(p_1^2\right)}$ and the following observation gives its normalized Laplacian spectrum.

Corollary 2

([13]). The normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, if $\mathfrak{n}=p_1^2$ is

$$\left\{\begin{array}{cc}0& {\displaystyle \frac{p_1-1}{p_1-2}}\\ 1& p_1-1\end{array}\right\}.$$

When we choose $\mathfrak{m}=2$ and $\mathfrak{n}=0$ according to Theorem 3, the conclusion can be derived.

Corollary 3

([13]). The normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, if $\mathfrak{n}=p_1^4$ is

$$\left\{\begin{array}{ccccc}0& 1& {\displaystyle \frac{p_1^3-1}{p_1^3-2}}& {\displaystyle \frac{p_1^2-1}{p_1^2-2}}& {\displaystyle \frac{2p_1^5-6p_1^3-2p_1^2+6\pm \sqrt{A}}{2(p_1^5-2p_1^3-2p_1^2+4)}}\\ & & & & \\ 1& p_1^3-p_1^2-1& p_1-2& p_1^2-p_1-1& 1\end{array}\right\},$$

where $A=4p_1^{10}-8p_1^9-7p_1^8+8p_1^7+24p_1^6-40p_1^4+8p_1^3+8p_1^2+4$.

By setting $\mathfrak{m}=\mathfrak{n}=1$ in Theorem 3, the subsequent outcome is as follows.

Corollary 4.

The normalized Laplacian spectrum of $\Gamma ({\mathbb{Z}}_{p_1^2{p}_2^2})$ is

$$\left\{\begin{array}{cccc}1& {\displaystyle \frac{p_1{p}_2-1}{p_1{p}_2-2}}& {\displaystyle \frac{p_1{p}_2^2-1}{p_1{p}_2^2-2}}& {\displaystyle \frac{p_1^2{p}_2-1}{p_1^2{p}_2-2}}\\ & & & \\ p_1{p}_2^2+p_1^2{p}_2-2p_1{p}_2-4& \varphi \left(p_1{p}_2\right)-1& \varphi \left(p_1\right)-1& \varphi \left(p_2\right)-1\end{array}\right\}.$$

The matrix below provides the other normalized eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^2{p}_2^2})$.

Proof. 

Assume that $\mathfrak{n}=p_1^2{p}_2^2$, where $p_1$ and $p_2$ denoting distinct prime numbers. The proper divisors of $\mathfrak{n}$ include $p_1,p_1^2,p_2,p_2^2,p_1{p}_2,p_1{p}_2^2$, and $p_1^2{p}_2$. Therefore, ${\delta }_{\mathfrak{n}}$ is the graph $G_7:p_2\sim p_1^2{p}_2\sim p_2^2\sim p_1^2\sim p_1{p}_2^2\sim p_1$ and $p_1^2{p}_2\sim p_1{p}_2\sim p_1{p}_2^2\sim p_1^2{p}_2$. Lemma 4 gives us

$$\begin{array}{cc}\hfill \Gamma \left({\mathbb{Z}}_{p_1^2{p}_2^2}\right)& ={\delta }_{p_1^2{p}_2^2}[\Gamma \left(A_{p_1}\right),\Gamma \left(A_{p_1^2}\right),\Gamma \left(A_{p_2}\right),\Gamma \left(A_{p_2^2}\right),\Gamma \left(A_{p_1{p}_2}\right),\Gamma \left(A_{p_1{p}_2^2}\right),\Gamma \left(A_{p_1^2{p}_2}\right)]\hfill \\ \hfill & ={\delta }_{p_1^2{p}_2^2}[{\overline{\mathcal{K}}}_{\varphi \left(p_1{p}_2^2\right)},{\overline{\mathcal{K}}}_{\varphi \left(p_2^2\right)},{\overline{\mathcal{K}}}_{\varphi \left(p_1^2{p}_2\right)},{\overline{\mathcal{K}}}_{\varphi \left(p_1^2\right)},{\mathcal{K}}_{\varphi \left(p_1{p}_2\right)},{\mathcal{K}}_{\varphi \left(p_1\right)},{\mathcal{K}}_{\varphi \left(p_2\right)}].\hfill \end{array}$$

As per the proper divisor sequence, we proceed to assign labels to the vertices contained within $G_7$, so that ${\mathfrak{n}}_1=\varphi \left(p_1{p}_2^2\right),{\mathfrak{n}}_2=\varphi \left(p_2^2\right),{\mathfrak{n}}_3=\varphi \left(p_1^2{p}_2\right),{\mathfrak{n}}_4=\varphi \left(p_1^2\right),{\mathfrak{n}}_5=\varphi \left(p_1{p}_2\right),{\mathfrak{n}}_6=\varphi \left(p_1\right)$, and ${\mathfrak{n}}_7=\varphi \left(p_2\right)$. Now, By Theorem 2, ${\displaystyle \frac{{\mathfrak{X}}_1}{s_1+{\mathfrak{X}}_1}}={\displaystyle \frac{{\mathfrak{X}}_2}{s_2+{\mathfrak{X}}_2}}={\displaystyle \frac{{\mathfrak{X}}_3}{s_3+{\mathfrak{X}}_3}}={\displaystyle \frac{{\mathfrak{X}}_4}{s_4+{\mathfrak{X}}_4}}=1,{\displaystyle \frac{{\mathfrak{X}}_5}{s_5+{\mathfrak{X}}_5}}={\displaystyle \frac{p_1+p_2-2}{p_1{p}_2-2}},{\displaystyle \frac{{\mathfrak{X}}_6}{s_6+{\mathfrak{X}}_6}}={\displaystyle \frac{p_1(p_2^2-1)}{p_1{p}_2^2-2}}$, and ${\displaystyle \frac{{\mathfrak{X}}_7}{s_7+{\mathfrak{X}}_7}}={\displaystyle \frac{p_2(p_1^2-1)}{p_1^2{p}_2-2}}$. Then, the eigenvalue 1 with multiplicity $\varphi \left(p_1{p}_2^2\right)-1+\varphi \left(p_2^2\right)-1+\varphi \left(p_1^2{p}_2\right)-1+\varphi \left(p_1^2\right)-1=p_1{p}_2^2+p_1^2{p}_2-2p_1{p}_2-4$ are the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{p_1^2{p}_2^2}\right)$, the eigenvalue ${\displaystyle \frac{p_1{p}_2-1}{p_1{p}_2-2}}$ with multiplicity $\varphi \left(p_1{p}_2\right)-1$, the eigenvalue ${\displaystyle \frac{p_1{p}_2^2-1}{p_1{p}_2^2-2}}$ with multiplicity $\varphi \left(p_1\right)-1$, the eigenvalue ${\displaystyle \frac{p_1^2{p}_2-1}{p_1^2{p}_2-2}}$ with multiplicity $\varphi \left(p_2\right)-1$, and the matrix below provides the remaining seven eigenvalues:

$$\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_1{\mathfrak{n}}_6}{{\mathfrak{X}}_{1}S}}}& 0\\ & & & & & & \\ 0& 1& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_2{\mathfrak{n}}_4}{{\mathfrak{X}}_1{\mathfrak{X}}_2}}}& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_2{\mathfrak{n}}_6}{{\mathfrak{X}}_{2}S}}}& 0\\ & & & & & & \\ 0& 0& 1& 0& 0& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_3{\mathfrak{n}}_7}{{\mathfrak{X}}_{1}T}}}\\ & & & & & & \\ 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_2{\mathfrak{n}}_4}{{\mathfrak{X}}_2{\mathfrak{X}}_4}}}& 0& 1& 0& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_4{\mathfrak{n}}_7}{{\mathfrak{X}}_{4}T}}}\\ & & & & & & \\ 0& 0& 0& 0& {\displaystyle \frac{{\mathfrak{X}}_5}{{\mathfrak{X}}_5+s_5}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_5{\mathfrak{n}}_6}{P}}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_5{\mathfrak{n}}_7}{Q}}}\\ & & & & & & \\ -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_1{\mathfrak{n}}_6}{{\mathfrak{X}}_{1}S}}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_2{\mathfrak{n}}_6}{{\mathfrak{X}}_{2}S}}}& 0& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_5{\mathfrak{n}}_6}{P}}}& {\displaystyle \frac{{\mathfrak{X}}_6}{{\mathfrak{X}}_6+s_6}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_6{\mathfrak{n}}_7}{R}}}\\ & & & & & & \\ 0& 0& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_3{\mathfrak{n}}_7}{{\mathfrak{X}}_{1}T}}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_4{\mathfrak{n}}_7}{{\mathfrak{X}}_{4}T}}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_5{\mathfrak{n}}_7}{Q}}}& -\sqrt{{\displaystyle \frac{{\mathfrak{n}}_6{\mathfrak{n}}_7}{R}}}& {\displaystyle \frac{{\mathfrak{X}}_7}{{\mathfrak{X}}_7+s_7}}\end{array}\right],$$

where $P=({\mathfrak{X}}_5+s_5)({\mathfrak{X}}_6+s_6),Q=({\mathfrak{X}}_5+s_5)({\mathfrak{X}}_7+s_7),R=({\mathfrak{X}}_6+s_6)({\mathfrak{X}}_7+s_7),S={\mathfrak{X}}_6+s_6$ and $T={\mathfrak{X}}_7+s_7$. □

Theorem 4.

The normalized Laplacian spectrum of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$, where $T_1=2\mathfrak{m}+1\le 2\mathfrak{n}+1=T_2$ consists of the eigenvalues

$$\{\begin{array}{ccccc}1& 1& 1& \dots & {\displaystyle \frac{p_1^{\mathfrak{m}+1}{p}_2^k-1}{p_1^{\mathfrak{m}+1}{p}_2^k-2}}\\ & & & & \\ \varphi \left(p_1^{T_1-i}{p}_2^{T_2}\right)-1& \varphi \left(p_1^{T_1}{p}_2^{T_2-x}\right)-1& \varphi \left(p_1^{T_1-1}{p}_2^{T_2-x}\right)-1& \dots & \varphi \left(p_1^{\mathfrak{m}}{p}_2^{T_2-x}\right)-1\end{array}$$

$$\begin{array}{cc}\hfill \cdots & {\displaystyle \frac{p_1^{2\mathfrak{m}+1}{p}_2^k-1}{p_1^{2\mathfrak{m}+1}{p}_2^k-2}}\hfill \\ \hfill \cdots & \varphi \left(p_2^{T_2-k}\right)-1\hfill \end{array}\},$$

where $i=1,2,\dots ,\mathfrak{m},\dots ,T_1,x=1,2,\dots ,T_2,k=1,2,\dots 2\mathfrak{n}.$ The matrix Equation (1) eigenvalues are the remaining normalized Laplacian eigenvalues of $\Gamma ({\mathbb{Z}}_{p_1^{T_1}{p}_2^{T_2}})$.

The previously stated outcome provides the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, where $\mathfrak{n}=p_1^{T_1}{p}_2^{T_2}$ and both $T_1$ and $T_2$ are odd. The demonstration for the previously mentioned result follows a similar approach to that outlined in Theorem 3.

Taking $T_2=0$ in Theorem 4, we obtain the following result.

Corollary 5.

Let $\mathfrak{n}=p_1^{2t+1}$, where $t\ge 2$ is a positive integer and $p_1>2$ is a prime. Then, the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ consists of the eigenvalue 1 with multiplicity $p_1^{2t}-p_1^t-t$, the eigenvalue ${\displaystyle \frac{p_1^i-1}{p_1^i-2}}$ with multiplicity $\varphi \left(p_1^{2t+1-i}\right)-1$, where $i=t+1,t+2,\dots ,2t-1,2t$. The matrix below provides the other normalized eigenvalues of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$:

$$\left[\begin{array}{cccccc}{X}_t& & & Y_{\left(t\right)\times \left(t\right)}& & \\ & y_{t+1}& x_{t+1,t+2}& \dots & x_{t+1,2t-1}& x_{t+1,2t}\\ & x_{t+2,t+1}& y_{t+2}& \dots & x_{t+2,2t-1}& x_{t+2,2t}\\ Y^T& ⋮& ⋮& \ddots & ⋮& ⋮\\ & x_{2t-1,t+1}& x_{2t-1,t+2}& \dots & y_{2t-1}& x_{2t-1,2t}\\ & x_{2t,t+1}& x_{2t,t+2}& \dots & x_{2t,2t-1}& y_{2t}\end{array}\right],$$

where $X_t=diag(1,1,\dots ,1,1)$,

$$Y=\left[\begin{array}{ccccc}0& 0& \dots & 0& x_{1,2t}\\ 0& 0& \dots & x_{2,2t-1}& x_{2,2t}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& x_{t-1,t+1}& \dots & x_{t-1,2t-1}& x_{t-1,2t}\\ x_{t,t+1}& x_{t,t+2}& \dots & x_{t,2t-1}& x_{t,2t}\end{array}\right]$$

and $x_{ij}=x_{ji}=-\sqrt{{\displaystyle \frac{{\mathfrak{n}}_i{\mathfrak{n}}_j}{({\mathfrak{X}}_i+s_i)({\mathfrak{X}}_j+s_j)}}}$, for $1\le i,j\le 2t$; $y_{ij}={\displaystyle \frac{s_i}{{\mathfrak{X}}_j+s_j}}$, for $i=t+1,t+2,\dots ,2t-1,2t$ and $s_i=\left\{\begin{array}{cc}0\hfill & fori=1,2,\dots ,t-1,t\hfill \\ \varphi \left(p_1^{2t+1-i}\right)-1\hfill & fori=t+1,t+2,\dots ,2t-1,2t.\hfill \end{array}\right.$

When $\mathfrak{m}=1$ and $p_2=1$ in Theorem 4, the subsequent outcome is obtained.

Corollary 6

([13]). The normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, if $\mathfrak{n}=p_1^3$ is

$$\left\{\begin{array}{cccc}0& 1& {\displaystyle \frac{p_1^2-1}{p_1^2-2}}& {\displaystyle \frac{2p_1^2-p_1-2}{p_1^2-2}}\\ & & & \\ 1& p_1^2-p_1-1& p_1-2& 1\end{array}\right\}.$$

When $\mathfrak{m}=0$ and $\mathfrak{n}=0$ in Theorem 4, the subsequent outcome is obtained.

Corollary 7

([13]). The normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$, if $\mathfrak{n}=p_1{p}_2$

$$\left\{\begin{array}{ccc}0& 1& 2\\ 1& p_1+p_2-4& 1\end{array}\right\}.$$

5. Conclusions and Future Work

In this study, we have explored the normalized Laplacian spectrum of the zero-divisor graph $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for a general class of $\mathfrak{n}$, where $\mathfrak{n}$ is expressed as $p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ being distinct primes and $T_1,T_2$ positive integers. By leveraging fundamental algebraic properties of the zero-divisor graph and detailed computations, we derived the normalized Laplacian eigenvalues for various configurations of $\mathfrak{n}$. This work extends previous results on specific classes of $\mathfrak{n}$, and demonstrates the interplay between the algebraic structure of ${\mathbb{Z}}_{\mathfrak{n}}$ and the spectral properties of its associated graph.

The findings reveal that the normalized Laplacian spectrum encapsulates significant information about the structural composition of zero-divisor graphs. In particular, the characteristic polynomials and eigenvalue distributions provide insights into the modular arithmetic and divisors underlying ${\mathbb{Z}}_{\mathfrak{n}}$. The results also validate that certain classes of zero-divisor graphs exhibit unique spectral patterns, reinforcing the importance of spectral graph theory in algebraic contexts.

This research opens several promising directions for future investigation. One potential avenue involves extending the spectral analysis to larger classes of finite commutative rings, with the aim of identifying deeper relationships between their algebraic properties and spectral parameters. Another area of interest is the exploration of additional graph invariants, such as the spectral radius, chromatic number, and connectivity, and their connections to the normalized Laplacian spectrum of zero-divisor graphs. Advanced computational techniques could also be employed to analyze the spectra for higher powers of primes and rings with multiple prime factors, potentially uncovering intricate patterns and properties. Furthermore, the spectral properties of zero-divisor graphs could have practical applications in cryptography, coding theory, and error detection frameworks, which warrant detailed investigations. A comparative study of the spectral characteristics of zero-divisor graphs with other algebraically defined graphs, such as unit graphs or co-maximal graphs, could reveal meaningful commonalities and distinctions. Additionally, future studies may consider works related to ring theory, such as “The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities” [16], which could provide further insights into the algebraic structures associated with zero-divisor graphs.

Collectively, this work not only enhances the theoretical understanding of zero-divisor graphs, but also establishes a robust foundation for interdisciplinary research integrating algebra, graph theory, and computational methodologies.