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The bipartite divisor graph B(X), for a set X of positive integers, and some of its properties have recently been studied. We construct the bipartite divisor graph for the product of subsets of positive integers and investigate some of its properties. We also give some applications in group theory.2010 Mathematics subject classification: primary 05C25; secondary 05C75.

Given a finite group G, the bipartite divisor graph for its conjugacy class sizes is the bipartite graph with bipartition consisting of the set of conjugacy class sizes of G \ Z(G) (where Z(G) denotes the centre of G) and the set of prime numbers that divide these conjugacy class sizes, and with {p, n} being an edge if gcd(p, n) = 1.In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K2,5, giving a solution to a question of Taeri [13].

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