Perfect Nilpotent Graphs

M. J. Nikmehr, A. Azadi

Let $R$ be a commutative ring with identity. The nilpotent graph of $R$, denoted by $\Gamma_N(R)$, is a graph with vertex set $Z_N(R)^*$, and two vertices $x$ and $y$ are adjacent if and only if $xy$ is nilpotent, where $Z_N(R)= \{x \in R \mid xy$ is nilpotent, for some $y \in R^*\}. $ A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose $\Gamma_N(R)$ is perfect. In addition, it is shown that for a ring $R$, if $R$ is Artinian, then $\omega(\Gamma_N(R))=\chi(\Gamma_N(R))=|{\rm Nil}(R)^*|+|{\rm Max}(R)|$.