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THE REGULAR GRAPH OF A NONCOMMUTATIVE RING

Published online by Cambridge University Press:  13 February 2013

S. AKBARI  and
F. HEYDARI
S. AKBARI*
Affiliation:
Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
F. HEYDARI
Affiliation:
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran email f-heydari@kiau.ac.ir
*
✉s_akbari@sharif.edu
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Abstract

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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 $R$, $\mathrm{Reg} (\Gamma (R))$, be the induced subgraph of $T(\Gamma (R))$ on the regular elements of $R$. In 2008, Anderson and Badawi proved that the girth of the total graph and the 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 $2\not\in Z(R)$, then the chromatic number and the clique number of $\mathrm{Reg} (\Gamma (R))$ 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 $\mathrm{Reg} (R)$ is finite, then $R$ is finite.

Keywords

regular graphtotal graphgirthchromatic number
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 132 - 140
Copyright
©2013 Australian Mathematical Publishing Association Inc. 

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