On the Diameter of Unitary Cayley Graphs of Rings

Su, Huadong

Canadian mathematical bulletin, Tome 59 (2016) no. 3, pp. 652-660

Résumé

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.

Détail

DOI : 10.4153/CMB-2016-014-7

Mots-clés : 05C25, 16U60, 05C12, unitary Cayley graph, diameter, k-good, unit sum number, self-injective ring