nilpotent graphs of skew polynomial rings over non-commutative rings

Let r be a ring and α 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 zn(r)={x∈r|xy isnilpotent, for some y∈r∗}. 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]) ‎. ‎it is shown that if r is a symmetric and α -compatible with exactly two minimal primes‎, ‎then diam(γn(r[x,α]))=2 ‎. ‎also we prove that γn(r) is a complete graph if and only if r is isomorphic to z2×z2 ‎.