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 $Z_N(R)={xin R;|; xy; rm{is; nilpotent,;for; some}; yin R^*}.$ 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 $Z_2timesZ_2$.