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On the Diameter of Unitary Cayley Graphs of Rings

Published online by Cambridge University Press:  20 November 2018

Huadong Su
Huadong Su*
Affiliation:
School of Mathematical and Statistics Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China and Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John's, Nfld A1C 5S7 e-mail: huadongsu@sohu.com
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Abstract

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The unitary Cayley graph of a ring $R$, denoted $\Gamma \left( R \right)$, is the simple graph defined on all elements of $R$, and where two vertices $x$ and $y$ are adjacent if and only if $x\,-\,y$ is a unit in $R$. The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$ and is denoted by $\text{diam}\left( G \right)$. It is proved that for each integer $n\,\ge \,1$, there exists a ring $R$ such that $\text{diam}\left( \Gamma \left( R \right) \right)=n$. We also show that $\text{diam}\left( \Gamma \left( R \right) \right)\in \left\{ 1,2,3,\infty \right\}$ for a ring $R$ with ${R}/{J\left( R \right)}\;$ self-injective and classify all those rings with $\text{diam}\left( \Gamma \left( R \right) \right)\,=\,1,\,2,\,3$, and $\infty$, respectively.

Keywords

05C2516U6005C12unitary Cayley graphdiameterk-goodunit sum numberself-injective ring
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 652 - 660
Copyright
Copyright © Canadian Mathematical Society 2016

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