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Abstract
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The prime vertex graph, ∆(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.
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