On the Common Divisor Graph of the Product of Integer Multisets

The common divisor graph, \(\Gamma (X)\) $\mathrm{\Gamma }(X)$ , 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, \(\Gamma (Z)\) $\mathrm{\Gamma }(Z)$ , when Z is a multiset of positive integers that admits a decomposition \(Z=XY\) $Z=XY$ , where \(XY=\{ xy | x\in X, y\in Y \}\) $XY=\{xy|x\in X,y\in Y\}$ and \(1\in X\) $1\in X$ and \(1 \in Y\) $1\in Y$ . Our results can be applied for the graphs associated to character degrees and conjugacy classes of the direct product of two finite groups.