Classifying Families of Character Degree Graphs of Solvable Groups

TL;DR

This paper characterizes which graphs formed by connecting two complete graphs can be realized as prime character degree graphs of solvable groups, advancing understanding of their structure.

Contribution

It provides a complete classification of the family of graphs _{k,t} as prime character degree graphs of solvable groups, identifying exactly which ones occur.

Findings

Identifies the conditions under which _{k,t} graphs occur as prime character degree graphs.

Provides a complete classification of these graphs for solvable groups.

Enhances understanding of the structure of prime character degree graphs.

Abstract

We investigate prime character degree graphs of solvable groups. In particular, we consider a family of graphs $\Gamma_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion. In this paper we determine completely which graphs $\Gamma_{k,t}$ occur as the prime character degree graph of a solvable group.