NILPOTENT GRAPHS OF SKEW POLYNOMIAL RINGS OVER NON-COMMUTATIVE RINGS

摘要

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 is an element of R vertical bar xy is nilpotent; for some y is an element of 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) x Z(2).