Classifying character degree graphs with seven vertices

TL;DR

This paper classifies which seven-vertex graphs can be prime character degree graphs of finite solvable groups, completing the classification for disconnected graphs and identifying specific connected graphs that occur.

Contribution

It provides a complete classification for disconnected graphs and identifies 22 connected graphs that can serve as prime character degree graphs.

Findings

22 connected graphs occur as prime character degree graphs

Two of these graphs have diameter three

Remaining graphs are constructed as direct products

Abstract

We study here the graphs with seven vertices in an effort to classify which of them appear as the prime character degree graphs of finite solvable groups. This classification is complete for the disconnected graphs. Of the 853 non-isomorphic connected graphs, we were able to demonstrate that twenty-two occur as prime character degree graphs. Two are of diameter three, while the remaining are constructed as direct products. Forty-four graphs remain unclassified.