On the Diameter of Unitary Cayley Graphs of Rings

Canadian mathematical bulletin

Huadong Su¹ 

[1] School of Mathematical and Statistics Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China

关键词: unitary Cayley graph;    diameter;    $k$-good;    unit sum number;    self-injective ring;   

DOI  :  10.4153/CMB-2016-014-7

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

The unitary Cayley graph of a ring $R$, denoted$Gamma(R)$, is the simple graphdefined 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 largestdistance between all pairs of vertices of a graph $G$ is calledthediameter of $G$, and is denoted by ${m diam}(G)$. It is provedthat for each integer $ngeq1$, there exists a ring $R$ suchthat${m diam}(Gamma(R))=n$. We also show that ${mdiam}(Gamma(R))in {1,2,3,infty}$ for a ring $R$ with $R/J(R)$self-injective and classify all those rings with ${mdiam}(Gamma(R))=1$, 2, 3 and $infty$, respectively.

【 授权许可】

Unknown