چکیده انگلیسی:
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 $x,y$, $xrightarrow y$ 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 $R[[x;alpha]]$ 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.
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