Nilpotent graphs of skew polynomial rings over non-commutative rings

Let R be a ring and α be a ring endomorphism of R. The undirected nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set ZN(R)∗, and two distinct vertices x and y are connected by an edge if and only if xy is nilpotent, where . In this article, we investigate the interplay between the ring theoretical properties of a skew polynomial ring R[x;α] and the graph-theoretical properties of its nilpotent graph ΓN(R[x;α])