Directed zero-divisor graph and skew power series rings

‎Let $R$ be an associative ring with identity and $Z^{\ast}(R)$ 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 $\Gamma{(R)}$‎, ‎is the directed graph whose vertices are the set of non-zero zero-divisors of $R$ and for distinct non-zero zero-divisors $x,y$‎, ‎$x\rightarrow y$ is an directed edge if and only if $xy=۰$‎. ‎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 $R[[x;\alpha]]$ and the graph-theoretical properties of its directed zero-divisor graph $\Gamma(R[[x;\alpha]])$‎. ‎In doing so‎, ‎we give a characterization of the possible diameters of $\Gamma(R[[x;\alpha]])$ in terms of the diameter of $\Gamma(R)$‎, ‎when the base ring $R$ is reversible and right Noetherian with an‎ ‎$\alpha$-condition‎, ‎namely $\alpha$-compatible property‎. ‎We also provide many examples for showing the necessity of our assumptions‎.

Keywords:

‎Zero-divisor graphs‎ , ‎Diameter‎ , ‎Reversible rings‎ , ‎Noetherian rings‎ , ‎Skew power series rings