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Classifying Prime Character Degree Graphs With Eight Vertices

Mark L. Lewis, Andrew Summers

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.

Classifying Prime Character Degree Graphs With Eight Vertices

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.
Paper Structure (14 sections, 12 theorems, 9 equations, 93 figures)

This paper contains 14 sections, 12 theorems, 9 equations, 93 figures.

Table of Contents

  1. Introduction
  2. Machinery
  3. Overall Results
  4. Graphs of Diameter Three
  5. Admissible Vertices and Families of Graphs
  6. Algorithm
  7. Disconnected Graphs
  8. Component Sizes 7 & 1
  9. Component Sizes 6 & 2
  10. Connected Graphs
  11. Graphs of Clique Sizes Seven & One
  12. Graphs of Clique Sizes Six & Two
  13. Graphs of Clique Sizes Five & Three
  14. Graphs of Clique Sizes Four & Four

Key Result

Theorem 2.1

Let $G$ be a solvable group and $\pi$ be a set of primes contained in $\Delta(G)$. If $|\pi|=3$, then there exists an irreducible character of $G$ with degree divisible by at least two of the primes from $\pi$. (In other words, any three vertices of the prime character degree graph of a solvable gro

Figures (93)

  • Figure 1: The graph classification algorithm
  • Figure 2: The two disconnected graphs which do not occur
  • Figure 3: The two disconnected graphs which occur
  • Figure 4: The occurring disconnected graph with component sizes seven and one
  • Figure 5: The occurring disconnected graph with component sizes six and two
  • ...and 88 more figures

Theorems & Definitions (14)

  • Theorem 2.1: Pálfy's Condition palfy1998character
  • Theorem 2.2: Theorem A of akhlaghi2018character
  • Corollary 2.3: Corollary B of akhlaghi2018character
  • Theorem 2.4: Theorem 3 of palfy2001character
  • Lemma 2.5: Theorems 2 and 4 of sass2016character
  • Definition 2.6: from bissler2025family
  • Lemma 2.7: Lemma 2.3 of bissler2025family
  • Theorem 2.8: Main Theorem of bissler2019classifyingfamilies
  • Theorem 2.9: Theorem 1.1 of laubacher2021prime
  • Theorem 2.10: Main Theorem of degroot2022prime
  • ...and 4 more