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

Abstract

Let $G$ be a finite group, and let $\mathrm{cd}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta (G)$ of $G$ is defined as the simple undirected graph whose vertex set $\mathrm{V}(G)$ consists of the prime divisors of the numbers in $\mathrm{cd}(G)$, two distinct vertices $p$ and $q$ being adjacent if and only if $pq$ divides some number in $\mathrm{cd}(G)$. In this note, we provide an upper bound on the size of $\mathrm{V}(G)$ in terms of the clique number $\omega (G)$ (i.e., the maximum size of a subset of $\mathrm{V}(G)$ inducing a complete subgraph) of $\Delta (G)$. Namely, we show that $|\mathrm{V}(G)|\le \max \{2\omega (G)+1,3\omega (G)-4\}$. Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3,4,9], and answers a question posed by the first author and H.P. Tong-Viet in [4].