Related papers: Eulerian of the Zero Divisor graph $\Gamma[\mathbb…

For a group $G$ and a subset $X$ of $G$, the commuting graph of $X$, denoted by $\Gamma(G,X)$ is the graph whose vertex set is $X$ and any two vertices $u$ and $v$ in $X$ are adjacent if and only if they commute in $G$. In this article,…

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes excluding~$1$. Let us define a directed graph $\Gamma(G)$, the set of vertices of this graph is $N(G)$ and the vertices $x$ and $y$ are connected by a directed…

The distinguishing number $D(\Gamma)$ of a graph $\Gamma$ is the least size of a partition of the vertices of $\Gamma$ such that no non-trivial automorphism of $\Gamma$ preserves this partition. We show that if the automorphism group of a…

The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK=G$. In this paper, we continue the study of $\Gamma(G)$, especially…

Using the new extension of the zero-divisor graph $\widetilde{\Gamma}(R)$ introduced in \cite{Groupe}, we give an approach of the diameter of $\Gamma(R)$ and $\Gamma(R[X])$ other than given in \cite{Lucas} thus we give a complete…

For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in…

A graph $G$ is called collapsible if for every even subset $R\subseteq V(G)$, there is a spanning connected subgraph $H$ of $G$ such that $R$ is the set of vertices of odd degree in $H$. A graph is the reduction of $G$ if it is obtained…

In this paper, we examine the linear codes with respect to the Hamming metric from incidence matrices of the zero-divisor graphs with vertex set is the set of all non-zero zero-divisors of the ring $\mathbb{Z}_n$ and two distinct vertices…

Let $R$ be a commutative ring with unity not equal to zero and let $\Gamma(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total perfect code} denoted by $C(G)$ in $G$ is a subset…

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if…

A retract of a graph $\Gamma$ is an induced subgraph $\Psi$ of $\Gamma$ such that there exists a homomorphism from $\Gamma$ to $\Psi$ whose restriction to $\Psi$ is the identity map. A graph is a core if it has no nontrivial retracts. In…

In this paper, we determine the structures of zero-divisor semigroups whose graph is $K_n + 1$, the complete graph $K_n$ together with an end vertex. We also present a formula to calculate the number of non-isomorphic zero-divisor…

The \textit{boxicity} (\textit{cubicity}) of an undirected graph $\Gamma$ is the smallest non-negative integer $k$ such that $\Gamma$ can be represented as the intersection graph of axis-parallel rectangular boxes (unit cubes) in…

Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$.…

We have tried to translate some graph properties of AG(R) and Gamma(R) to the topological properties of Zariski topology. We prove that Rad(Gamma(R)) and Rad(AG(R)) are equal and they are equal to 3, if and only if the zero ideal of R is an…

Let $R$ be a commutative ring with identity and $\Bbb A (R)$ be the set of ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$, denoted by $A_{I} (R) $, is a simple graph with the vertex set $\Bbb A(R)^{\ast} := \Bbb…

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a…

Let X be a non-empty finite set and alpha a symmetric bilinear form on a real finite dimensional vector space E. We say that a set GG={U_i | i in X} of linear lines in E is an isometric sheaf, if there exist generators u_i of the lines U_i,…

For a nonabelian group G, the non-commuting graph $\Gamma_G$ of $G$ is defined as the graph with vertex set $G-Z(G)$, where $Z(G)$ is the center of $G$, and two distinct vertices of $\Gamma_G$ are adjacent if they do not commute in $G$. In…

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…