Directed zero-divisor graph and skew power series rings

‎Let R be an associative ring with identity and Z∗(R) be its set of non-zero zero-divisors‎. ‎Zero-divisor graphs of rings are well represented in the literature of commutative and non-commutative rings‎. ‎The directed zero-divisor graph of R‎, ‎denoted by Γ(R)‎, ‎is the directed graph whose vertices are the set of non-zero zero-divisors of R and for distinct non-zero zero-divisors x,y‎, ‎x→y is an directed edge if and only if xy=0‎. ‎In this paper‎, ‎we connect some graph-theoretic concepts with algebraic notions‎, ‎and investigate the interplay between the ring-theoretical properties of a skew power series ring R[[x;α]] and the graph-theoretical properties of its directed zero-divisor graph Γ(R[[x;α]])‎. ‎In doing so‎, ‎we give a characterization of the possible diameters of Γ(R[[x;α]]) in terms of the diameter of Γ(R)‎, ‎when the base ring R is reversible and right Noetherian with an‎ ‎α-condition‎, ‎namely α-compatible property‎. ‎We also provide many examples for showing the necessity of our assumptions