Groups whose Bipartite Divisor Graph for Character Degrees Has Five Vertices

Let G be a finite group and cd∗ (G) be the set of nonlinear irreducible character degrees of G. Suppose that ρ(G) denotes the set of primes dividing some element of cd∗ (G). The bipartite divisor graph for the set of character degrees which is denoted by B(G), is a bipartite graph whose vertices are the disjoint union of ρ(G) and cd∗ (G), and a vertex p ∈ ρ(G) is connected to a vertex a ∈ cd∗ (G) if and only if p|a. In this paper, we investigate the structure of a group G whose graph B(G) has five vertices. Especially we show that all these groups are solvable.