On the number of components of a graph related to character degrees

Abstract

ABSTRACT. We connect two nonlinear irreducible character of a finite group G if their degrees have a common prime divisor. In this paper we show that the corresponding graph has at most three connected components.

Zeros of Brauer characters and the defect zero graph

“…Now remove those primes from D 0 ðGÞ which divide no character degree. It has been shown by Manz, Staszewski and Willems [15] that this latter graph has at most three connected components. (In fact they consider a kind of dual graph GðGÞ, with vertices the degrees of non-linear characters, and edges given by common prime divisors, but this clearly has the same number of connected components.…”

Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

“…We now assume S = A n , where n 11. We consider the character χ r,s of the symmetric group S n corresponding to the partition λ r,s = (n − s − r, s + 1, 1 r−1 ) of n. (See §3 of [13] for notation.) This character of S n exists whenever r 1, s 0, and n r + 2s + 1, and has degree…”

K4-free character graphs with seven vertices

“…A useful way to study the character degree set of a finite group G is to associate a graph to cd(G). One of these graphs is the character graph ∆(G) of G [12]. Its vertex set is ρ(G) and two vertices p and q are joined by an edge if the product pq divides some character degree of G. We refer the readers to a survey by Lewis [8] for results concerning this graph and related topics.…”