Groups whose character degree graph has diameter three

摘要

Let G be a finite group, and let Delta(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. It is well known that, whenever Delta(G) is connected, the diameter of Delta(G) is at most 3. In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M. L. Lewis.