Related papers: Rings whose total graphs have genus at most one

The non-commuting graph $\Gamma_R$ of a finite ring $R$ with center $Z(R)$ is a simple undirected graph whose vertex set is $R \setminus Z(R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab \ne ba$. In this paper, we…

In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if…

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and…

Let $R$ be a commutative ring and let $U(R)$ be multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, $\Gamma(R, G, S)$, corresponding to a multiplicative subgroup…

Let $R$ be a commutative ring with identity. Let $\Gamma(R)$ be a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. In this paper we consider a subgraph $\Gamma_2(R)$ of…

Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we…

Let $R$ be a commutative ring with nonzero identity and $I$ a proper ideal of $R$. The {\it ideal-based zero-divisor graph} of $R$ with respect to the ideal $I$, denoted by $\Gamma_I(R)$, is the graph on vertices $\{x \in R\setminus I \mid…

Suppose that $R$ is a finite commutative ring with identity. The involutory Cayley graph $\G(R)$ of $R$ is the graph whose vertices are the elements of $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $(x-y)^2=1$. In…

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…

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…

Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $\Gamma(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of…

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The annihilator graph of $R$ is defined as the undirected graph $AG(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct…

Let $R$ be a finite product of integral domains and $D$ be a union of prime ideals (it is possible that $R$ is just an integral domain). Let $n \geq 1$ be a positive integer. This paper introduces the $n$-total graph of a $(R, D)$. The…

For a commutative ring $R$ with identity, the zero-divisor graph of $R$, denoted $\Gamma(R)$, is the graph whose vertices are the non-zero zero divisors of $R$ with two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In…

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such…

In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\,…

Let $R$ be a commutative ring with identity. We define a graph $\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in \aut$ such that…

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by $\Gamma_R^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $[x,y] \neq r$ and $-r$. In…

A graph is an instrument which is extensively utilized to model various problems in different fields. Up to date, many graphs have been developed to represent algebraic structures, particularly rings in order to study their properties. In…

Assume that $R$ is a commutative ring with nonzero identity. In this paper, we introduce and investigate zero-annihilator graph of $R$ denoted by $\mathtt{ZA}(R)$. It is the graph whose vertex set is the set of all nonzero nonunit elements…