Finite groups with non-complete character codegree graphs

Abstract

The codegree graph $\Gamma \left(G\right)$ of a finite group G is defined as follows: its vertices consist of all prime divisors of $\left|G\right|$, and distinct vertices p and q are connected by an edge if and only if G admits an irreducible character with codegree divisible by pq. In this paper, we describe the finite groups whose codegree graphs are not complete, and in particular, we provide a complete description of nonsolvable groups whose codegree graphs have diameter three. Additionally, we show that $\Gamma \left(H\right)$ is a subgraph of $\Gamma \left(G\right)$ whenever H is a subgroup of G.