Five-vertex degree graphs of nonsolvable groups

Abstract

The prime character degree graph Δ ( G ) of a finite group G is the graph whose vertices are all prime divisors of the degrees of the complex irreducible characters of G , with distinct primes p and q joined by an edge if pq divides χ ( 1 ) for some complex irreducible character χ of G . Lewis and White determined all graphs with four vertices that occur as Δ ( G ) for some nonsolvable group G . In this paper, we determine all graphs with five vertices- up to two exceptions that occur as Δ ( G ) for some nonsolvable group G . Along with previously known results on prime character degree graphs of solvable groups, this completes the classification of all five-vertex graphs- up to two exceptions that occur as Δ ( G ) for some finite group G .