相关论文: Coloring in essential annihilating-ideal graphs of…

Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$. Let $R\Join I$ be the subring of $R\times R$ consisting of the elements $(r,r+i)$ for $r\in R$ and $i\in I$. We study the diameter and girth of the zero-divisor…

Two subgraphs $A,B$ of a graph $G$ are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if $G$ is a graph with bounded clique number, and sufficiently large chromatic number, then it has two…

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in…

The {\em chromatic edge-stability number} $es_{\chi}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a spanning subgraph with the chromatic number smaller than that of $G$. A graph $G$ is called {\em…

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple…

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…

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\,:\,…

A strong edge-coloring of a graph $G$ is a coloring of edges of $G$ such that every color class forms an induced matching. The strong chromatic index is the minimum number of colors needed to color the graph. The Ore-degree $\theta(G)$ of a…

Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The…

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…

An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

Fr\"oberg's classical theorem about edge ideals with $2$-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have…

A cover of an associative (not necessarily commutative nor unital) ring $R$ is a collection of proper subrings of $R$ whose set-theoretic union equals $R$. If such a cover exists, then the covering number $\sigma(R)$ of $R$ is the…

Three edges $e_{1}, e_{2}$ and $e_{3}$ in a graph $G$ are consecutive if they form a path (in this order) or a cycle of length three. An injective edge coloring of a graph $G = (V,E)$ is a coloring $c$ of the edges of $G$ such that if…

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we introduce the cozero-divisor graph $\acute{\Gamma}_I(R)$ of $R$ and obtain some related results.

Let G = (V, E) be a multigraph without loops and for any x {\in}V let E(x) be the set of edges of G incident to x. A homogeneous edge-coloring of G is an assignment of an integer m >= 2 and a coloring c:E {\to} S of the edges of…

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $Z(M)$ be the set of all zero-divisors on $M$. In 2008, D.F. Anderson and A. Badawi introduced the regular graph of $R$. In this paper, we generalize the regular graph of $R$…

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set ${{a,b}:a,b\in R, a-b\in R^\times}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is…

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…

In this paper, we characterize all graphs $G$ satisfying \[\operatorname{reg}(S/J_G)=\ell(G)=c(G)\] where $\ell(G)$ is the sum of the lengths of the longest induced paths in each connected component of $G$ and $c(G)$ is the number of the…