Zero Divisor Graphs of Skew Generalized Power Series Rings

Let R be a ring, (S,<TEX>≤</TEX>) a strictly ordered monoid and <TEX>ω</TEX> : S <TEX>→</TEX> End(R) a monoid homomorphism. The skew generalized power series ring R[[S,<TEX>ω</TEX>]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,<TEX>ω</TEX>]] and the graph-theoretical properties of its zero-divisor graph <TEX>Γ</TEX>(R[[S,<TEX>ω</TEX>]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.