A_γ Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings

Throughout this study, we discuss only undirected, connected, finite, and simple graphs. A graph G is represented by the pair $G\left(V\right(G),E(G\left)\right)$, whereas $V\left(G\right)$ and $E\left(G\right)$ represent the node and the edge sets of G, respectively. The size and order of G are, respectively, the cardinalities of $E\left(G\right)$ and $V\left(G\right)$. The degree of $w\in V\left(G\right)$ is indicated by $d_G\left(w\right)$ (or simply by $d_w$) and is equal to the number of edges incident on w. The neighborhood $N\left(w\right)$ of $w\in V\left(G\right)$ is the collection of nodes of G connected to w, so $d_w$ is the same as $\left|N\right(w\left)\right|$. If each node is of same degree, then G is said to be regular.

Consider the diagonal matrix $D\left(G\right)=diag(d_{v_1},d_{v_2},\dots ,d_{v_n})$ of node degrees $d_{v_i}$ of G, when $i=1,2,\dots ,n$. The adjacency matrix $A\left(G\right)={\left(a_{jk}\right)}_{n\times n}$ is a real symmetric matrix, where $\left(jk\right)$-th entry is 1, if $v_j$ is connected to $v_k$ and 0 otherwise. The matrices $Q\left(G\right)=D\left(G\right)+A\left(G\right)$ and $L\left(G\right)=D\left(G\right)-A\left(G\right)$ are, respectively, the signless Laplacian as well as the Laplacian matrices of G. Their multiset of eigenvalues is the signless Laplacian and the Laplacian spectrums of G, respectively. The Laplacian and the signless Laplacian are positive real semi-definite matrices, so their spectrum is real, and they are ordered as ${\mathsf{\lambda }}_n\left(G\right)\le {\mathsf{\lambda }}_{n-1}\left(G\right)\le \cdots \le {\mathsf{\lambda }}_1\left(G\right)$ and ${\mu }_n\left(G\right)\le {\mu }_{n-1}\left(G\right)\le \cdots \le {\mu }_1\left(G\right)$, respectively. Further details about these matrices can be seen in [1,2].

Nikiforov [3] suggested to investigate the symmetrical configurations $A_{\gamma }\left(G\right)$ of $D\left(G\right)$ and $A\left(G\right)$, and it is specified as $A_{\gamma }\left(G\right)=\gamma D\left(G\right)+(1-\gamma )A\left(G\right)$, whereas $1\ge \gamma \ge 0$. Certainly, $A\left(G\right)=A_0\left(G\right)$, $D\left(G\right)=A_1\left(G\right)$ and $Q\left(G\right)=A\left(G\right)+D\left(G\right)=2A_{\frac{1}{2}}\left(G\right)$. Thus, $A_{\gamma }\left(G\right)$ is a generalization of $A\left(G\right)$ as well as $Q\left(G\right)$ of G. Due to the fact that $A_{\gamma }\left(G\right)$ is symmetric and real, so its eigenvalues are ordered as ${\mathsf{\lambda }}_1\left(A_{\gamma }\left(G\right)\right)\ge {\mathsf{\lambda }}_2\left(A_{\gamma }\left(G\right)\right)\ge \cdots \ge {\mathsf{\lambda }}_n\left(A_{\gamma }\left(G\right)\right),$ whenever ${\mathsf{\lambda }}_1\left(A_{\gamma }\left(G\right)\right)$ is referred to as the generalized adjacency spectral radius of G. Moreover, $A_{\gamma }\left(G\right)$ ($\gamma \ne 1$) is irreducible and non-negative for connected graph G. As a result, ${\mathsf{\lambda }}_1\left(A_{\gamma }\left(G\right)\right)$ is a simple eigenvalue (Perron–Frobenius theorem), and its associated eigenvector Y with positive entries is the generalized adjacency Perron vector of G. The spectral properties of $A_{\gamma }\left(G\right)$ are described in [3,4,5,6,7] and the references listed therein.

Consider a commutative ring R, with multiplicative identity $1\ne 0$. If there exists $x_2\in R$ ($x_2\ne 0$) such that $x_1{x}_2=0$, then $x_1\in R$ ($x_1\ne 0$) is referred to as a zero divisor of $R.$ The collection of zero divisors is symbolized by $Z\left(R\right)$, while $Z\left(R\right)\setminus \left\{0\right\}=Z^{\ast }\left(R\right)$ is the collection of non-zero zero divisors of R. The zero divisor graph $\Gamma \left(R\right)$ of R is a graph, where $Z^{\ast }\left(R\right)$ is its node set and two different nodes $y,z\in Z^{\ast }\left(R\right)$ are connected whenever $yz=zy=0$. Beck [8] established such graphs over commutative rings; in his concept, he incorporated the identity and was primarily concerned with the coloring of commutative rings. Following that, the authors of [9] updated the concept of $\Gamma \left(R\right)$ by omitting the identity of R. The finite field of order n is represented by ${\mathbb{F}}_n$ and a ring of integers modulo n by ${\mathbb{Z}}_n$. The order of $\Gamma \left({\mathbb{Z}}_n\right)$ is $n-1-\varphi \left(n\right)$, whereas $\varphi$ is Euler’s phi function. The graph theoretic characteristics of $\Gamma \left({\mathbb{Z}}_n\right)$ are widely investigated [10,11].

In [12], the authors showed that $\Gamma \left({\mathbb{Z}}_n\right)$ is a Laplacian integral, where n is some prime power. According to [13], whenever a connected graph is bipartite, its lowest signless Laplacian eigenvalue equals 0, and the multiplicity of the eigenvalue 0 equals the number of bipartite components. Afkhami et al. [14] defined the normalized Laplacian as well as the signless Laplacian spectrums of $\Gamma \left({\mathbb{Z}}_n\right)$ and evaluated such spectra over a range of n values. In addition, they have identified certain bounds for various eigenvalues of the normalized and signless Laplacian matrices of $\Gamma \left({\mathbb{Z}}_n\right)$. In [15], the authors examined the adjacency spectrum of $\Gamma \left({\mathbb{Z}}_n\right)$. Furthermore, the normalized (signless) Laplacian eigenvalues were discussed in [14,16,17,18,19,20,21,22,23,24] carried out the Laplacian and the adjacency spectral analysis. We apply the standard, symbol $K_n$ for the complete graph, ${\overline{K}}_n$ as its complement, and $K_{a,b}$ for the complete bipartite graph. Additional unexplained terminologies and notations may be found in [25].

We have investigated many articles for the spectral graph theory to learn in depth the applications and use of chemical substances. From the applications point of view, the use of the eigenvalues and especially in the Laplacian matrix plays a vital role in the computer algorithms, where they play a foundational role in machine learning. In addition, it can also be used for load balancing in in these algorithms. In computers, nowadays, the image processing is very important for the security point as well as other archaeological points of view. In these processes, the adjacency matrix plays a key role in the visualization and other zooming purposes. In addition, there is a build up of a strong inter-network connection for certain topologies that the algebraic graph theory can play in such a circumstances. The connections inside the super computers are based on certain topologies, and its working rule is based on famous Cayley graphs that use the concept of symmetry.

There are still significant gaps in the existing work about the identification of certain $A_{\gamma }$ eigenvalues of zero divisor graphs for commutative and von Neumann rings. The apparent reason for this is that neither the construction of zero divisor graphs over rings is well specified nor it is feasible to derive convenient formulas of graph characteristics for wide classes of rings. We make an attempt in this article to examine one of these problems.

The remaining article is organized as follows: Section 2 begins with some fundamental findings that will be employed to compute the $A_{\gamma }$ eigenvalues of $\Gamma \left({\mathbb{Z}}_n\right)$. Section 3 discusses the $A_{\gamma }$ eigenvalues of zero divisor graphs over von Neumann regular rings. Section 4 contains the paper’s conclusion and future work.

We begin this section with a couple of definitions and well-known outcomes, which we use to demonstrate our main results.

Consider the $n\times n$ matrix such that the columns of M and rows M are partitioned as per the partition $P=\{{\pi }_1,\cdots ,{\pi }_l\}$ of the $\pi =\{1,2,\cdots ,l\}$ set. The matrix $\mathcal{Q}$ is an $l\times l$ matrix with entries equal to the mean column or rows sum of the $t_{i,j}$ blocks of M; such matrix is known as the quotient matrix, see [1,26]. If every $t_{i,j}$ has a fixed column (row) sum, the P is known as regular, and $\mathcal{Q}$ is said to be the regular quotient matrix. For generality, the eigenvalues of $\mathcal{Q}$ and M are the same.

Let $G_i$ be regular graphs, the subsequent result from [27] indicates the $A_{\gamma }$ spectrum of the joined union of $G_i$ in relation with the adjacency spectrum of $G_i$, for $n\ge i\ge 1$, together with the eigenvalues of $\mathcal{Q}$.

Assume that ${{\rm Y}}_n$ is the simple connected graph, where $d_1,\dots ,d_s$ of n is a set of proper divisors with two distinct nodes being adjacent whenever n divides $d_i{d}_j.$

For $s\ge i\ge 1$, consider where $(z,n)$ represents the G.C.D of z and n. Notice that $C_{d_i}\cap C_{d_j}=\varphi$, when $j\ne i$; this implies that $C_{d_1},C_{d_2},\dots ,C_{d_s}$ are disjoint and partitions $V\left(\Gamma \left({\mathbb{Z}}_n\right)\right)$ as: According to the $C_{d_i}$ definition, a node of $C_{d_i}$ is edge connected to the node of $C_{d_j}$ in $\Gamma \left({\mathbb{Z}}_n\right)$ when n divides $d_i{d}_j$, where $j,i\in \{1,2,\dots ,s\}$. In addition, the order of $C_{d_i}$ is $\varphi \left(\frac{n}{d_i}\right)$, where $s\ge i\ge 1,$ (see [22]).

The succeeding lemma presents important properties about the subgraphs that are either null graphs or cliques.

The sequel results give the structure of $\Gamma \left({\mathbb{Z}}_n\right)$.

Furthermore, we examine the $A_{\gamma }$ eigenvalues of $\Gamma \left({\mathbb{Z}}_n\right)$ of ${\mathbb{Z}}_n$.

Next, we discuss the $A_{\gamma }$ spectrum for some special classes of zero divisor graph. As a reminder, $K_n$ has an adjacency spectrum $\{n-1,{(-1)}^{[n-1]}\}$ and that of ${\overline{K}}_n$ is $\{0^{\left[n\right]}\}$.

Following the steps as in Theorem 3, we can prove the odd case.

The next result gives the $A_{\gamma }$ eigenvalues of $\Gamma \left({\mathbb{Z}}_n\right)$, where n is the multiplication of $three$ prime numbers.

By putting $\gamma =0$ and $\gamma =\frac{1}{2}$ in Theorem 2 and its consequences, we obtain the adjacency spectrum while the signless Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_n\right)$ is obtained in [14,17,20]. Similarly, using the fact $(\gamma -\beta )L\left(G\right)=(\gamma -\beta )(D\left(G\right)-A\left(G\right))=A_{\gamma }\left(G\right)-A_{\beta }\left(G\right)$ and Theorem 2 along with its consequences, we obtain the Laplacian eigenvalues, which were earlier obtained in [12,21].

A ring R is known as von Neumann regular if there exists $z\in R$ so that $y=y^{2}z$ for each $y\in R$. The collection of idempotents of R is represented by $B\left(R\right),$ and its zero divisor graph is represented by $\Gamma \left(B\right(R\left)\right)$. In [28,29,30,31,32], researchers examined the zero divisor graphs of von Neumann regular rings and the adjacency spectrum was recently given in [33].

If $r_1\in R$, then the annihilator of $r_1$ is denoted by $Ann\left(r_1\right)$ and is defined as $Ann\left(r_1\right)=\{r_2\in R:r_1{r}_2=0\}$. Define a relation $r_1\sim r_2$ on $R,$ if $Ann\left(r_1\right)=Ann\left(r_2\right)$ and ∼ is clearly an equivalence relation. In [29], the authors show the graph isomorphism, and the equivalence class has a particular idempotent if R is a von Neumann regular. Patil and Shinde [33] proved that for every non-trivial idempotent, the equivalence class of e has an independent subgraph and two nodes $a,b\in \Gamma \left(R\right)$ are edge connected whenever $e_a$ and $e_b$ are edge connected in $\Gamma \left(B\right(R\left)\right)$. They also showed that for a non-trivial idempotent $e=(e_1,e_2,\dots ,e_k)$ in ${\mathbb{F}}_1\times {\mathbb{F}}_2\times \cdots \times {\mathbb{F}}_k$, the cardinality of $A_e$ is ${\displaystyle \prod _{e_{i\ne 0}}}\left(\right|{\mathbb{F}}_i|-1),$ where × is the usual product of rings (fields).

The structure of $\Gamma \left(R\right)$ of the von Neumann regular rings R is obtained by the following result.

Now, we discuss the $A_{\gamma }$ eigenvalues of $R.$

If $R\cong {\mathbb{F}}_1\times {\mathbb{F}}_2\times \cdots \times {\mathbb{F}}_k$, for every ${\mathbb{F}}_i\cong {\mathbb{Z}}_{p_i}$ and $p_i,i=1,\dots ,k$ are distinct primes, the $A_{\gamma }$ spectrum of $\Gamma \left(R\right)$ is presented by Corollary 1. Thus, by Theorem 5, we may determine the spectrum of more general classes of zero divisors graphs of rings.

Next, we discuss some consequence of Theorem 5. First, we will find the $A_{\gamma }$ spectrum of ${\mathbb{F}}_1\times {\mathbb{F}}_2$, whereas ${\mathbb{F}}_1$ and ${\mathbb{F}}_2$ are finite fields. If ${\mathbb{F}}_1\cong {\mathbb{Z}}_p$ and ${\mathbb{F}}_1\cong {\mathbb{Z}}_q$, whereas $p<q$ are primes, then $A_{\gamma }$ eigenvalues are as in Lemma 5; otherwise, the $A_{\gamma }$ spectrum is presented by the sequel result.