On regular bipartite divisor graph for the set of irreducible character degrees

Abstract

Given a finite group G, the bipartite divisor graph, denoted by B(G), for its irreducible character degrees is the bipartite graph with bipartition consisting of cd(G)*, where cd(G)* denotes the nonidentity irreducible character degrees of G and the p(G) which is the set of prime numbers that divide these degrees, and with {p, n} being an edge if gcd(p, n) not equal 1. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where B(G) is a path or a cycle and discussed some properties of G. In particular she proved that B(G) is a cycle if and only if G is solvable and B(G) is either a cycle of length four or six. Inspired by 2-regularity of cycles, in this paper we consider the case where B(G) is an n-regular graph for n is an element of {1,2,3}. In particular we prove that there is no solvable group whose bipartite divisor graph is C-4 + C-6.