$K_4$-free character graphs with seven vertices

Abstract

For a finite group $G$, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we determine the structure of all finite groups $G$ with $K_4$-free character graph $\Delta (G)$ having seven vertices. We also obtain a classification of all $K_4$-free graphs with seven vertices which can occur as character graphs of some finite groups.