On Graphs Associated with Character Degrees and Conjugacy Class Sizes of Direct Products of Finite Groups

Abstract

Abstract.The prime vertex graph, Δ(X), and the common divisor graph, Γ(X), are two graphs that have been deûned on a set of positive integers X. Some properties of these graphs have been studied in the cases where either X is the set of character degrees of a group or X is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.