On the character degree graph of finite groups

Abstract

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G), two distinct vertices p and q being adjacent if and only if pq divides some number in cd(G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of [1], where the solvable case was treated.

Bounding the number of vertices in the degree graph of a finite group

“…As mentioned in the Introduction, the next result (which depends on Theorem A of [2]) will be crucial in the proof of the main result of this paper. In the following statement, by ∆(G) we denote the complement of the graph ∆(G): this is the graph whose vertex set is the same as of ∆(G), and two vertices are adjacent in ∆(G) if and only if they are not adjacent in ∆(G).…”

“…A key ingredient in our proof of Theorem A is the main theorem of [2], which provides some detailed structural information concerning finite groups G such that |V(G)| ≤ 2ω(G) does not hold. The part of this information that is relevant for our purposes is gathered in Proposition 2.4.…”

“…Finally, in the following discussion every group is assumed to be finite, and the classification of finite simple groups is involved (via the main theorem of [2]).…”

K4-free character graphs with seven vertices

“…Using Lemma 2.1, χ(1)/θ(1) divides [G : L]. Thus as G/L is a π ′ -group, xu|θ (1). It is a contradiction as x and u are non-adjacent vertices in ∆(L).…”

“…If θ extends to P , then as all Sylow subgroups of G/P are cyclic, θ is extendible to G. Thus we assume that θ does not extend to P . Hence using Lemma 2.1, all character degrees in cd(P|θ) are divisible by pθ (1). Now let χ ∈ Irr(G|θ).…”