SOLVABLE GROUPS WITH CHARACTER DEGREE GRAPHS HAVING 5 VERTICES AND DIAMETER 3

ABSTRACT

Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees in cd(G), and there is an edge between p and q if pq divides some degree in cd(G). In this paper, we show that if Δ(G) has 5 vertices, then the diameter of Δ(G) is at most 2. This shows that the example in^[9] of a solvable group G where Δ(G) has diameter 3 has the fewest number of vertices possible.