  Let $G$ be a finite group, and let ${\rm{cd}}(G)$ 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 ${\rm{V}}(G)$
consists of the prime divisors of the numbers in ${\rm{cd}}(G)$, two distinct
vertices $p$ and $q$ being adjacent if and only if $pq$ divides some number in
${\rm{cd}}(G)$. In this note, we provide an upper bound on the size of
${\rm{V}}(G)$ in terms of the clique number $\omega(G)$ (i.e., the maximum size
of a subset of ${\rm{V}}(G)$ inducing a complete subgraph) of $\Delta(G)$.
Namely, we show that
$|{\rm{V}}(G)|\leq{\rm{max}}\{2\omega(G)+1,\;3\omega(G)-4\}$. 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].

Group Theory