$K_4$-free character graphs with seven vertices

TL;DR

This paper classifies all finite groups with seven-vertex character graphs that do not contain a complete subgraph of four vertices, providing a comprehensive understanding of their structure and possible graph configurations.

Contribution

It determines the structure of all finite groups with $K_4$-free seven-vertex character graphs and classifies all such graphs realizable as character graphs.

Findings

Complete classification of groups with $K_4$-free seven-vertex character graphs

Identification of all possible $K_4$-free graphs as character graphs

Structural descriptions of these groups and graphs

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.