The regular graph of a noncommutative ring

S. Akbari; F. Heydari

The regular graph of a noncommutative ring

Authors

S. Akbari Sharif University of Technology
F. Heydari

Keywords:

Regular graph, Total graph, Girth, Chromatic number

Abstract

Let

$R$ be a ring and

$Z(R)$ be the set of all zero-divisors of

$R$ . The total graph of

$R$ , denoted by

$T(\Gamma(R))$ is a graph with all elements of

$R$ as vertices, and two distinct vertices

$x,y\in R$ are adjacent if and only if

$x+y\in Z(R)$ . Let the regular graph of

$R$ ,

$Reg(\Gamma(R))$ , be the induced subgraph of

$T(\Gamma(R))$ on the regular elements of

$R$ . In 2008, Anderson and Badawi proved that the girth of total graph and regular graph of a commutative ring are contained in the set

$\{3,4,\infty\}$ . In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if

$R$ is a reduced left Noetherian ring and

$2\notin Z(R)$ , then the chromatic number and the clique number of

$Reg(\Gamma(R))$ are the same and they are

$2^r$ , where

$r$ is the number of minimal prime ideals of

$R$ . Among other results we show that if

$R$ is a semiprime left Noetherian ring and

$Reg(R)$ is finite, then

$R$ is finite. 10.1017/S0004972712001177

Published

2013-12-02

Issue

Vol. 89 No. 1 (2014)

Section

Articles

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