On r-noncommuting graph of finite rings

TL;DR

This paper investigates the properties of the r-noncommuting graph of finite rings, revealing its irregularity and characterizing the ring structures through subgraph analysis.

Contribution

It introduces and analyzes the r-noncommuting graph of finite rings, providing new insights into its structure and characterizations of the underlying rings.

Findings

The graph is not regular, a lollipop, or complete bipartite.

Characterizations of rings based on induced subgraphs.

Insights into the structure of finite rings via graph properties.

Abstract

Let $R$ be a finite ring and $r\in R$. The $r$-noncommuting graph of $R$, denoted by $\Gamma_R^r$, is a simple undirected graph whose vertex set is $R$ and two vertices $x$ and $y$ are adjacent if and only if $[x,y] \neq r$ and $-r$. In this paper, we study several properties of $\Gamma_R^r$. We show that $\Gamma_R^r$ is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of $\Gamma_R^r$ (induced by the non-central elements of $R$) and obtained some characterizations of $R$.