[Paper Review] A family of graphs that cannot occur as character degree graphs of solvable groups

Mark W. Bissler, Mark L. Lewis+1 more|arXiv (Cornell University)|Jul 10, 2017

Finite Group Theory Research9 references7 citations

TL;DR

This paper establishes a new family of graphs that cannot occur as the character degree graph Δ(G) of any finite solvable group. Using techniques based on Pálfy’s condition, admissibility of vertices, and induction on group structure, the authors prove that if a graph has two adjacent vertices of degree two with no common neighbor, it cannot arise as Δ(G) for any solvable group G.

ABSTRACT

We investigate character degree graphs of solvable groups. In particular, we provide general results that can be used to eliminate which degree graphs can occur as solvable groups. Finally, we show a specific family of graphs cannot occur as a character degree for any solvable group.

Motivation & Objective

• To identify structural constraints that prevent certain graphs from being character degree graphs of solvable groups.
• To extend existing results on character degree graphs by introducing new elimination criteria based on vertex degree and adjacency.
• To prove that a specific family of graphs—characterized by two adjacent degree-two vertices with no common neighbor—cannot occur as Δ(G) for any solvable group.
• To develop and apply a framework of admissible vertices and normal subgroup reduction to eliminate candidate graphs.
• To generalize earlier results on character degree graphs by incorporating Pálfy’s condition and results from [10] and [11] on diameter-three graphs.

Proposed method

• Define a vertex as admissible if its removal (and incident edges) results in a subgraph that cannot be a character degree graph of any solvable group.
• Use Lemma 2.1 to show that if a prime p is admissible, then O^p(G) = G, implying no normal p-complement exists.
• Apply Lemma 2.2 to show that if all vertices in a graph are admissible, the graph cannot be a character degree graph.
• Use Lemma 2.3 to rule out normal Sylow q-subgroups under specific adjacency and non-adjacency conditions.
• Employ induction and subgroup reduction via normal subgroups H and G/H to analyze possible character degree sets.
• Leverage results from [5], [9], and [10] on central Sylow subgroups and diameter-three graphs to derive contradictions when candidate graphs arise.

Experimental results

Research questions

• RQ1 Under what conditions can a graph fail to be the character degree graph of a solvable group?
• RQ2 Can a graph with two adjacent degree-two vertices and no common neighbor occur as Δ(G) for a solvable group G?
• RQ3 How do admissibility and Pálfy’s condition interact to eliminate candidate graphs?
• RQ4 What structural constraints arise from the presence of strongly admissible vertices in a character degree graph?
• RQ5 Can the absence of normal Sylow subgroups be used to rule out entire families of graphs as character degree graphs?

Key findings

• A family of graphs with k ≥ 5 vertices, two adjacent degree-two vertices with no common neighbor, and satisfying Pálfy’s condition cannot be character degree graphs of any solvable group.
• The proof relies on showing that such graphs lead to contradictions via normal subgroup reduction and the existence of central Sylow subgroups in quotients.
• For each k ≥ 5, there exist exactly two such graphs (up to isomorphism) satisfying the hypotheses, and neither can occur as Δ(G) for any solvable G.
• The authors establish that if every vertex in a graph is admissible, then the graph cannot be a character degree graph of any solvable group.
• The method successfully eliminates graphs with diameter three and specific vertex degree patterns using results from [5], [9], and [10].
• The contradiction arises when assuming such a graph occurs as Δ(G), leading to the existence of a proper normal subgroup H with O^p(G) < G, violating admissibility of p.