Related papers: Compressed Intersection Annihilator Graph
Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $\Gamma^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct…
The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. We introduce commutatively closed graphs and investigate properties of commutatively…
In this article we study the annihilating-ideal graph of the ring $C(X)$. We have tried to associate the graph properties of $\mathbb{AG}(X)$, the ring properties of $C(X)$ and the topological properties of $X$. We have shown that $ X $ has…
The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the…
In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$…
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…
Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if the…
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 an artinian ring $R$, the directed zero-divisor graph $\Gamma(R)$ is connected if and only if there is no proper one-sided identity element in $R$. Sinks and sources are characterized and clarified for finite ring $R$, especially, it is…
We prove a generalized version of Evans and Griffith's Improved New Intersection Theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and…
This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of $\Om(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract $\G_r(R)$ of $\G(R)$, where $R$ is a commutative ring. We show that the core of…
In this paper we study zero--divisor graphs of rings and semirings. We show that all zero--divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic…
Let $(A, \oplus, *, 0)$ be an MV-algebra, $(A, \odot, 0)$ be the associated commutative semigroup, and $I$ be an ideal of $A$. Define the ideal-based zero-divisor graph $\Gamma_{I}(A)$ of $A$ with respect to $I$ to be a simple graph with…
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 finite commutative ring with $1\ne 0$. In this article, we study the total graph of $R$, denoted by $\tau (R)$, determine some of its basic graph-theoretical properties, determine when it is Eulerian, and find some conditions…
For $a\in R$, let $P_a$ denote the intersection of all minimal prime ideals of $R$ containing $a$. An ideal $I$ of a ring $R$ is called a $z^{\circ}$-ideal if $P_a\subseteq I$ for all $a\in I$. In this paper, we first investigate the class…
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 commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…
Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be a simple graph, an $L(2,1)$-labeling of $\mathcal{G}$ is an assignment of labels from nonnegative integers to vertices of $\mathcal{G}$ such that adjacent vertices get labels which differ 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…