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This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two…

In this paper, we study non-zero component graph $\Gamma(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ over a finite field $\mathbb{F}$. We show that the graph is Hamiltonian and not Eulerian. We also characterize the…

For a commutative ring $R$, the zero-divisor graph of $R$ is a simple graph with the vertex set as the set of all zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. This article attempts to…

Inspired by the work in \cite{sauer} regarding the classification of all the zero-divisor graphs with six vertices, we obtain all the zero-divisor graphs with seven vertices. Hence we classify all the zero-divisor commutative semigroups…

In this article, explicit formulas for finding the determining number and the metric dimension of the zero-divisor graph of Z_n and non-Boolean semisimple rings are given. In the case of Boolean rings, an upper bound of the determining…

The zero divisor graph of a commutative ring $R$ with unity is a graph whose vertices are the nonzero zero-divisors of the ring, with two distinct vertices being adjacent if their product is zero. This graph is denoted by $\Gamma(R)$. In…

In this article, we study the zero-divisor graph of the commutative non-chain ring with identity $ \mathbb{F}_p + u\mathbb{F}_p + v\mathbb{F}_p + uv\mathbb{F}_p,$ where \(u^2 = 0\), \(v^2 = 0\), \(uv = vu\), and \(p\) is an odd prime. We…

In this paper we study sub-semigroups of a zero-divisor semigroup $S$ determined by properties of the zero-divisor graph $\Gamma(S)$. We use these sub-semigroups to study the correspondence between zero- divisor semigroups and zero-divisor…

In this paper, we study the genera of zero-divisor graphs with respect to ideals in finite rings.

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…

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…

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…

In this paper, the generalized blow-up of a Boolean lattice $L\cong \textbf{2}^n$ using finite chains is introduced. Also, we compute the strong metric dimension of the zero-divisor graph of the blow-up of a Boolean lattice. These results…

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…

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. Chemical graph theory is a branch of mathematical…

The zero-divisor graph $\Gamma(R)$ of a ring $R$ is a graph with nonzero zero-divisors of $R$ as vertices and distinct vertices $x,y$ are adjacent if $xy=0$ or $yx=0$. We provide an equivalence relation on a ring $R$ and express $\Gamma(R)$…

This paper introduces a new approach to associating a graph with a commutative ring. Let $R$ be a commutative ring with identity. The unit-zero divisor graph of a commutative ring $R$, denoted by $G_{UZ}(R)$, offers a novel framework for…

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. In this paper we derive the Vertex and Edge…

For a ring $R$, the zero-divisor graph is a simple graph $\Gamma(R)$ whose vertex set is the set of all non-zero zero-divisors in a ring $R$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$ or $yx=0$ in $R$. By…

For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the…