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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