On the Connectivity of Non-Commuting Graph of Finite Rings

Discrete and Computational Geometry, Graphs, and Games

Abstract

The non-commuting graph of a non-commutative ring $R$, denoted by $\Gamma_{R}$, is a simple graph with vertex set of elements in $R$ except for its center. Two distinct vertices $x$ and $y$ are adjacent if $xy \neq yx$. In this paper, we study the vertex-connectivity and edge-connectivity of a non-commuting graph associated with a finite non-commutative ring $R$ and prove their lower bounds. We show that the edge-connectivity of $\Gamma_{R}$ is equal to its minimum degree. The vertex-connectivity and edge-connectivity of $\Gamma_{R}$ are determined when $R$ is a non-commutative ring of order $p^{n}$ where $p$ is a prime number, and $n \in \left\{2,3,4,5\right\}$.