Related papers: Laplacian spectrum of weakly zero-divisor graph of…

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$, denoted by $\Gamma'(R)$, is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$, and two distinct vertices $x$ and $y$ are…

For a commutative ring $R$ with identity, the \emph{weakly zero-divisor graph} $W\Gamma(R)$ has vertex set $Z(R)^{\ast}$, with distinct vertices $x$ and $y$ adjacent whenever there exist nonzero $r\in{\rm Ann}(x)$ and $s\in{\rm Ann}(y)$…

The weakly zero-divisor graph $W\Gamma(R)$ of a commutative ring $R$ is the simple undirected graph whose vertices are nonzero zero-divisors of $R$ and two distinct vertices $x$, $y$ are adjacent if and only if there exist $w\in {\rm…

The zero-divisor graph $\Gamma(R)$ of a ring $R$ is a graph with nonzero zero-divisors of $R$ as vertices and distinct vertices $x,y$ are adjacent if $xy=0$ or $yx=0$. We provide an equivalence relation on a ring $R$ and express $\Gamma(R)$…

The cozero-divisor graph of a commutative ring $R$, denoted $\Gamma'(R)$, is the graph whose vertices are the non-zero and non-unit elements of $R$, with two distinct vertices $x$ and $y$ adjacent if and only if $x \notin Ry$ and $y \notin…

Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two…

We study the Laplacian eigenvalues of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$ of the ring $\mathbb{Z}_{n}$ and prove that $\Gamma\left(\mathbb{Z}_{p^t}\right)$ is Laplacian integral for every prime $p$ and positive…

For a ring $R$, the zero-divisor graph is a simple graph $\Gamma(R)$ whose vertex set is the set of all non-zero zero-divisors in a ring $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$ or $yx=0$ in $R$. By…

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. We consider the zero divisor graph…

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper, we consider the zero divisor graph…

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper we derive the Vertex and Edge…

Let $R$ be a ring with unity. The cozero-divisor graph of a ring $R$, denoted by $\Gamma'(R)$, is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of $R$, and two distinct vertices $x$ and $y$ are…

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of R and two vertices are adjacent if their product is zero. The compressed zero divisor graph $\Gamma_E[R]$ is…

Let R be a finite commutative ring with unity, and let G = (V, E) be a simple graph. The zero-divisor graph, denoted by {\Gamma}(R) is a simple graph with vertex set as R, and two vertices x, y \in R are adjacent in {\Gamma}(R) if and only…

In this article, we study the zero-divisor graph of the commutative non-chain ring with identity $ \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where \(u^2 = 0\), \(v^2 = 0\), \(uv = vu\), and \(p\) is an odd prime. We…

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. Chemical graph theory is a branch of mathematical…

The zero-divisor graph $\Gamma(R)$ of an associative ring $R$ is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of $R$, and two distinct vertices $x$ and $y$ are joined by an edge iff either $xy=0$ or…

For a commutative ring $R,$ with non-zero zero divisors $Z^{\ast}(R)$. The zero divisor graph $\Gamma(R)$ is a simple graph with vertex set $Z^{\ast}(R)$, and two distinct vertices $x,y\in V(\Gamma(R))$ are adjacent if and only if $x\cdot…

Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by $\Gamma_I(R)$, is the graph whose vertices are the set $\{x \in R \setminus I | xy \in I$ for some $y \in R…

We investigate eigenvalues of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$ and study the interplay between these eigenvalues, the ring-theoretic properties of $R$ and the graph-theoretic properties of $\Gamma(R)$. The…