Related papers: The Unit-Zero Divisor Graph of a Commutative Ring

Let $R$ be a commutative ring with $1\neq 0$ and $\Bbb{A}(R)$ be the set of ideals with nonzero annihilators. The annihilating-ideal graph of $R$ is defined as the graph $\Bbb{AG}(R)$ with the vertex set $\Bbb{A}(R)^{*} =…

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…

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

In this article, we have defined nil clean graph of a ring $R$. The vertex set is the ring $R$, two ring elements $a$ and $b$ are adjacent if and only if $a + b$ is nil clean in $R$. Graph theoretic properties like girth, dominating set,…

Let R be a commutative ring and M be an R-module. We define the large sum graph, denoted by \acute{G}(M), as a graph with the vertex set of non-large submodules of M and two distinct vertices are adjacent if and only if N + K is a non-large…

We determine the metric dimension of the zero-divisor graph of the matrix semiring over a commutative entire antinegative semiring.

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…

In this article we introduce the zero-divisor graphs $\Gamma_\mathscr{P}(X)$ and $\Gamma^\mathscr{P}_\infty(X)$ of the two rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$; here $\mathscr{P}$ is an ideal of closed sets in $X$ and…

A mixed graph $G$ is a graph obtained from a simple undirected graph by orientating a subset of edges. $G$ is self-converse if it is isomorphic to the graph obtained from $G$ by reversing each directed edge. For two mixed graphs $G$ and $H$…

The unitary Cayley graph $C_R$ of a finite unital ring $R$ is the simple graph with vertex set $R$ in which two elements $x$ and $y$ are connected by an edge if and only if $x-y$ is a unit of $R$. We characterize the unitary Cayley graph…

Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational…

Let $R$ be a commutative ring with identity. The involutory Cayley graph $\mathcal{G}(R)$ of $R$ is defined as the graph whose vertex set is the set of elements of $R$, where two vertices $a$ and $b$ are adjacent exactly when $(a-b)^2=1$.…

Let $R$ be a commutative ring with a non-zero identity and $\mathfrak{J}_R$ be its Jacobson graph. We show that if $R$ and $R'$ are finite commutative rings, then $\mathfrak{J}_R\cong\mathfrak{J}_{R'}$ if and only if $|J(R)|=|J(R')|$ and…

Let $R$ be a ring (not necessarily a commutative ring) with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph with vertices in the form of an ordered pair $(e,u)$, where $e$ is an idempotent and $u$ is a unit of ring $R$,…

Let $m>1$ be an integer, and let $I(\mathbb{Z}_m)^*$ be the set of all non-zero proper ideals of $\mathbb{Z}_m$. The intersection graph of ideals of $\mathbb{Z}_m$, denoted by $G(\mathbb{Z}_m)$, is a graph with vertices $I(\mathbb{Z}_m)^*$…

In this paper, we investigate the Sombor index of the total graph and unit graph of $\mathbb{Z}_n$ which is denoted by $T_{\Gamma}(\mathbb{Z}_n)$ and $G(\mathbb{Z}_n)$ respectively for $n \in \{2k, p^{\alpha}, pq, p^2q\}$ where $p$ and $q$…

Gcd-graphs represent an interesting and historically important class of integral graphs. Since the pioneering work of Klotz and Sander, numerous incarnations of these graphs have been explored in the literature. In this article, we define…

The paper is devoted to studying the orthogonality graph of the matrix ring over a commutative ring. It is proved that the orthogonality graph of the ring of matrices with size greater than 1 over a commutative ring with zero-divisors is…

Suppose a compact torus $T$ acts on a closed smooth manifold $M$. Under certain conditions, Guillemin and Zara associate to $(M, T)$ a labeled graph $\mG_M$ where the labels lie in $H^2(BT)$. They also define the subring $H_T^*(\mG_M)$ of…

In this paper, we are motivated by two conjectures proposed by C. Bender et al.\ in 2024, which have remained open questions. The first conjecture states that if the complemented zero-divisor graph \( G(S) \) of a commutative semigroup \( S…