Groups having complete bipartite divisor graphs for their conjugacy class sizes

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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\setminus\mathbf Z (G) (where \mathbf 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)\neq 1 . In this paper we construct infinitely many groups whose bipartite divisor graph for their conjugacy class sizes is the complete bipartite graph K_{2,5} , giving a solution to a question of Taeri [15].