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On the Diameter of Unitary Cayley Graphs of Rings

  Published:2016-06-07
 Printed: Sep 2016
  • Huadong Su,
    School of Mathematical and Statistics Sciences, Guangxi Teachers Education University, Nanning, Guangxi, 530023, P. R. China
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Abstract

The unitary Cayley graph of a ring $R$, denoted $\Gamma(R)$, 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 ${\rm diam}(G)$. It is proved that for each integer $n\geq1$, there exists a ring $R$ such that ${\rm diam}(\Gamma(R))=n$. We also show that ${\rm diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with ${\rm diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.
Keywords: unitary Cayley graph, diameter, $k$-good, unit sum number, self-injective ring unitary Cayley graph, diameter, $k$-good, unit sum number, self-injective ring
MSC Classifications: 05C25, 16U60, 05C12 show english descriptions Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]
Units, groups of units
Distance in graphs
05C25 - Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]
16U60 - Units, groups of units
05C12 - Distance in graphs
 

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