Eulerian character degree graphs of solvable groups

Abstract

Let G be a finite group, let Irr(G) be the set of all complex irreducible characters of G and let cd(G) be the set of all degrees of characters in Irr ( G ) . Let ρ ( G ) be the set of all primes that divide some degrees in cd ( G ) . The character degree graph Δ ( G ) of G is the simple undirected graph with vertex set ρ ( G ) and in which two distinct vertices p and q are adjacent if there exists a character degree r ∈ cd ( G ) such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph Δ ( G ) of a solvable group G to be eulerian. We also prove that every n – 2 regular graph on n vertices where n is even and n ≥ 4 is the character degree graph of some solvable group. In the last part of the paper we give a bound for the number of Eulerian character degree graphs in terms of number of vertices.