On the Common Divisor Graph of the Product of Integer Multisets.

The common divisor graph, Γ (X) , is a graph that has been defined on a set of positive integers X. Some properties of this graph have been studied in the cases when either X is the set of character degrees or the set of conjugacy class sizes of a group. This paper deals with several properties of a particular case of the common divisor graph, Γ (Z) , when Z is a multiset of positive integers that admits a decomposition Z = X Y , where X Y = { x y | x ∈ X , y ∈ Y } and 1 ∈ X and 1 ∈ Y . Our results can be applied for the graphs associated to character degrees and conjugacy classes of the direct product of two finite groups. [ABSTRACT FROM AUTHOR]