
On the Diameter of Unitary Cayley Graphs of Rings
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 $xy$ 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 $R/J(R)$
selfinjective 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, selfinjective ring Categories:05C25, 16U60, 05C12 