On prime character degree graphs occurring within a family of graphs

TL;DR

This paper studies certain connected graphs formed by two complete graphs connected in a specific way, determining which can occur as prime character degree graphs of solvable groups, and discusses potential generalizations.

Contribution

It characterizes which graphs formed by two complete graphs can be prime character degree graphs of solvable groups and explores possible generalizations.

Findings

Identifies specific graph structures that can occur as prime character degree graphs.

Provides criteria to determine the occurrence of these graphs in solvable groups.

Discusses potential extensions to broader classes of graphs.

Abstract

In this paper we investigate families of connected graphs which do not contain an odd cycle in their complement. Specifically, we consider graphs formed by two complete graphs connected in a particular way. We determine which of these graphs can or cannot occur as the prime character degree graph of a solvable group. An obvious expansion and generalization can also be considered, of which we make mention.