MMN-1475
The induced subgraph of the unitary Cayley graph of a commutative ring over regular elements
Ali Reza Naghipour;Abstract
Let $R$ be a finite commutative ring and $R^*$ be the group of
units of $R$. The unitary Cayley graph of $R$, denoted by $G(R)$,
is the graph obtained by setting all the elements of $R$ to be
the vertices and defining vertices $x$ and $y$ to be adjacent if
and only if $x-y\in R^{*}$. We denote by ${\Gamma}(R)$ the
induced subgraph of $G(R)$ whose vertex set is $R^{*}$. In this
paper, the basic properties of ${\Gamma}(R)$ are investigated
and some characterization results regarding connectedness,
chromatic number, chromatic index, girth and planarity of
${\Gamma}(R)$ are given. In addition, by using the concept of
spectrum of graphs, it is shown that for two finite commutative
rings $R$ and $S$, if ${\Gamma}(R)$ is non empty, then
${\Gamma}(R)\cong {\Gamma}(S)$ as graphs if and only if
$R/J(R)\cong S/J(R)$ as rings, where $J(R)$ and $J(S)$ are the
Jacobson radicals of $R$ and $S$, respectively. As a consequence
of this result, we obtain that the structure of a reduced ring
$R$ can be determined by ${\Gamma}(R)$.
Vol. 17 (2016), No. 2, pp. 965-977
DOI: 10.18514/MMN.2017.1475