Related papers: A matrix ring with commuting graph of maximal diam…

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and…

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…

Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in…

The commuting variety of matrices over a given field is a well-studied object in linear algebra and algebraic geometry. As a set, it consists of all pairs of square matrices with entries in that field that commute with one another. In this…

Let $G$ be a finite $p$-separable group, for some fixed prime $p$. Let $\Gamma_p(G)$ be the common divisor graph built on the set of non-central conjugacy classes of $p$-regular elements of $G$: this is the graph whose vertices are the…

The conditional diameter of a connected graph $\Gamma=(V,E)$ is defined as follows: given a property ${\cal P}$ of a pair $(\Gamma_1, \Gamma_2)$ of subgraphs of $\Gamma$, the so-called \emph{conditional diameter} or ${\cal P}$-{\em…

Let $R$ be a ring with unity. The upper ideal relation graph $\Gamma_U(R)$ of the ring $R$ is a simple undirected graph whose vertex set is the set of all non-unit elements of $R$ and two distinct vertices $x, y$ are adjacent if and only if…

Graph maps that are homotopic to the identity and that permute the vertices are studied. Given a periodic point for such a map, a {\em rotation element} is defined in terms of the fundamental group. A number of results are proved about the…

Structures of commuting semigroups of isometries under certain additional assumptions like double commutativity or dual double commutativity are found.

Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph {\Gamma}G (called non-commuting graph of G) as follows: Take G\Z(G) as the vertices of {\Gamma}G and join x and y, whenever $xy \not= yx$. In this paper, we…

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…

A fractional matching of a graph $G$ is a function $f:E(G) \to [0,1]$ such that for any $v\in V(G)$, $\sum_{e\in E_G(v)}f(e)\leq 1$ where $E_G(v) = \{e \in E(G): e$ is incident with $v$ in $G\}$. The fractional matching number of $G$ is…

In this paper we study the realizability question for commuting graphs of finite groups: Given an undirected graph $X$ is it the commuting graph of a group $G$? And if so, to determine such a group. We seek efficient algorithms for this…

Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by…

Let $S$ be a semigroup and $\mathbb F$ be a field. For an ideal $J$ of the semigroup algebra ${\mathbb F}[S]$ of $S$ over $\mathbb F$, let $\varrho _J$ denote the restriction (to $S$) of the congruence on ${\mathbb F}[S]$ defined by the…

The intersection graph $\Delta_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were…

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph $\Gamma_G$ associated to $G$ is the graph whose vertex set is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only if $xy\neq yx$.…

Efficiency of routing on a regular digraph often involves finding opitmal properties of the graph. For example, the diameter of a digraph is the maximum distance between any two vertices. We show how we can study these problems…

The orbital diameter of a primitive permutation group is the maximal diameter of its orbital graphs. There has been a lot of interest in bounds for the orbital diameter. In this paper we provide explicit bounds on the diameters of groups of…

The flip graph is the graph whose nodes correspond to non-isomorphic combinatorial triangulations and whose edges connect pairs of triangulations that can be obtained one from the other by flipping a single edge. In this note we show that…