Classifying Prime Character Degree Graphs With Eight Vertices

TL;DR

This paper classifies prime character degree graphs with eight vertices for finite solvable groups, identifying which graphs occur, do not occur, or remain unknown, using a new algorithm.

Contribution

It develops a general algorithm for classifying prime character degree graphs of any order and applies it to the case of eight vertices.

Findings

Classified 1,229 disconnected graphs of order eight.

Identified 37 connected graphs that occur, with 34 constructed via direct products.

Determined 56 graphs do not occur, with 206 graphs remaining unclassified.

Abstract

In this paper, an effort is made to classify which prime character degree graphs having eight vertices occur for some finite solvable group. To approach this, we compile known results and constructions from the literature which are used to develop a general algorithm to begin classifying graphs of any order. We then apply the algorithm to the graphs of order eight. Of the 12,346 non-isomorphic graphs with eight vertices, 1,229 are disconnected and are fully classified. Meanwhile, 37 of the 11,117 non-isomorphic connected graphs are shown to occur; 34 of which are constructed via direct products and 3 of which have diameter three. Fifty-six graphs are shown not to occur, several of which fall into previously studied families, while the classification of 206 graphs is still unknown.