2013/07/05 by Hung P. Tong-Viet, Tong-Viet, Hung P.
Mathematics · #05C25 #20C15 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.CO #math.GR #math.RT #msc:05C25 #msc:20C15
paper · pdf · doi:10.48550/arxiv.1307.2175
18 pages
arxiv created 2013/08/23 · arxiv updated 2013/08/27
Let G be a finite group and let Irr(G) be the set of all irreducible complex characters of G. Let cd(G) be the set of all character degrees of G and denote by ρ(G) the set of primes which divide some character degrees of G. The prime graph Δ(G) associated to G is a graph whose vertex set is ρ(G) and there is an edge between two distinct primes p and q if and only if the product pq divides some character degree of G. In this paper, we show that the prime graph Δ(G) of a finite group G is 3-regular if and only if it is a complete graph with four vertices.