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 [Formula: see text] Let [Formula: see text] be the set of all primes that divide some degrees in [Formula: see text] The character degree graph [Formula: see text] of G is the simple undirected graph with vertex set [Formula: see text] and in which two distinct vertices p and q are adjacent if there exists a character degree [Formula: see text] such that r is divisible by the product pq. In this paper we obtain a necessary condition for the character degree graph [Formula: see text] 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 [Formula: see text] 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.

Keywords

• Character graph
• Eulerian graph
• regular graph
• finite solvable group
• 05C45
• 20C15