Related papers: Exploring Ring Structures: Multiset Dimension Anal…

Let $R$ be a commutative ring with identity. We define a graph $\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in \aut$ such that…

If a graph $G_M$ is embedded into a closed surface $S$ such that $S \backslash G_M$ is a collection of disjoint open discs, then $M=(G_M,S)$ is called a {\em map}. A {\em zigzag} in a map $M$ is a closed path which alternates choosing, at…

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric…

This article studies the zero divisor graphs of the ring of Lipschitz integers modulo $n$. In particular we focus on the number of vertices, the diameter and the girth. We also give some results regarding the domination number of these…

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of…

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…

In this paper extremal values of the difference between several graph invariants related to the metric dimension are studied: mixed metric dimension, edge metric dimension and strong metric dimension. These non-trivial extremal values are…

Belk and Connelly introduced the realizable dimension $\textrm{rd}(G)$ of a finite graph $G$, which is the minimum nonnegative integer $d$ such that every framework $(G,p)$ in any dimension admits a framework in $\mathbb{R}^d$ with the same…

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…

Let $G=(V,E)$ be a connected graph, let $v\in V$ be a vertex and let $e=uw\in E$ be an edge. The distance between the vertex $v$ and the edge $e$ is given by $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A vertex $w\in V$ distinguishes two edges…

Zero-divisor graphs of commutative rings are well-represented in the literature. In this paper, we consider dominating sets, total dominating sets, domination numbers and total domination numbers of zero-divisor graphs. We determine the…

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph's metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This…

Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with…

Let $G$ be a graph, a dominating induced matching (DIM) of $G$ is an induced matching that dominates every edge of $G$. In this paper we show that if a graph $G$ has a DIM, then $\chi(G) \leqslant 3$. Also, it is shown that if $G$ is a…

In this paper, we study the zero divisor graph $\Gamma^m(L)$ of a multiplicative lattice L. We prove under certain conditions that for a reduced multiplicative lattice L having more than two minimal prime elements, $\Gamma^m(L)$ contains a…

In this paper, we study the strong zero-divisor graph of a p.q.-Baer $*$-ring. We determine the condition on a p.q.-Baer $*$-ring (in terms of the smallest central projection in a lattice of central projections of a $*$-ring), so that its…

Let $R$ be a commutative ring and $\Gamma(R)$ denote its zero-divisor graph. In this paper, we investigate the genus number of the compact Riemann surface which $\Gamma(R)$ can be embedded and illustrate all finite commutative rings $R$ (up…

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is…

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…

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…