On prime character degree graphs occurring within a family of graphs (ii)

TL;DR

This paper advances the classification of prime character degree graphs within a specific family, identifying which graphs correspond to solvable groups, with only two exceptions, by refining previous methods.

Contribution

It improves the classification technique for prime character degree graphs, narrowing down which graphs occur in solvable groups with minimal exceptions.

Findings

Most graphs in the family are realizable as prime character degree graphs of solvable groups.

Two notable graphs remain unclassified as realizable within the group context.

Refined approach enhances the understanding of the relationship between graphs and group characters.

Abstract

In this paper, we continue the classification work done in the first paper of the same name. With careful modifications of our previous approach, we are able to deduce (with two notable exceptions) which members of the previously introduced graph family manifest as the prime character degree graph of some solvable group.