Nilpotent graphs of skew polynomial rings over non-commutative rings

Abstract

Let $R$ be a ring and $\alpha$ be a ring endomorphism of $R$‎. ‎The undirected nilpotent graph of $R$‎, ‎denoted by $\Gamma_N(R)$‎, ‎is a graph with vertex set $Z_N(R)^*$‎, ‎and two distinct vertices $x$ and $y$ are connected by an edge if and only if $xy$ is nilpotent‎, ‎where $Z_N(R)=\{x\in R\;|\; xy\; \rm{is\; nilpotent,\;for\; some}\; y\in R^*\}.$ In this article‎, ‎we investigate the interplay between the ring theoretical properties of a skew polynomial ring $R[x;\alpha]$ and the graph-theoretical properties of its nilpotent graph $\Gamma_N(R[x;\alpha])$‎. ‎It is shown that if $R$ is a symmetric and $\alpha$-compatible with exactly two minimal primes‎, ‎then $diam(\Gamma_N(R[x,\alpha]))=2$‎. ‎Also we prove that $\Gamma_N(R)$ is a complete graph if and only if $R$ is isomorphic to $\Z_2\times\Z_2$‎.