An overview on the bipartite divisor graph for the set of irreducible character degrees

Let G be a finite group. The bipartite divisor graph B.G/ for the set of irreducible complex character degrees cd.G/ is the undirected graph with vertex set consisting of the prime numbers dividing some element of cd.G/ and of the nonidentity character degrees in cd.G/, where a prime number p is declared to be adjacent to a character degree m if and only if p divides m. The graph B.G/ is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: The prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to B.G/ by outlining the main results that have been proved so far, see for instance [10; 11; 25; 26; 27]. In this process we improve some of these results.