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arxiv: 2606.05331 · v1 · pith:SIRX6RHJnew · submitted 2026-06-03 · 🧮 math.GR

On prime character degree graphs occurring within a family of graphs (iii)

Mark W. Bissler , Thatcher Debowski , Theodore F. Hoelker , Jacob Laubacher , Lorenzo Ravaglia , G. Sivanesan This is my paper

Pith reviewed 2026-06-28 03:34 UTC · model grok-4.3

classification 🧮 math.GR
keywords prime character degree graphsolvable groupfinite groupcharacter degreegraph classification
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The pith

Adding flexibility to a graph construction completes the classification of prime character degree graphs arising from solvable groups.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper finishes a three-part classification of which graphs can appear as the prime character degree graph of a solvable group. Earlier papers handled restricted cases of a certain family of graphs. The present work introduces additional flexibility so that the construction covers the graphs in full generality. The authors then determine exactly which members of this broader family occur as prime character degree graphs of solvable groups and which do not.

Core claim

We conclude the classification work done in the two previous papers of the same name. Here we add flexibility to the construction, thereby viewing the graphs in full generality. Our goal, as ever, is to determine which graphs do or do not occur as the prime character degree graph of a solvable group.

What carries the argument

The flexible construction of graphs in an extended family that models prime character degree graphs of solvable groups.

If this is right

  • Every prime character degree graph of a solvable group now belongs to the extended family.
  • Certain graphs inside the family are shown not to arise from any solvable group.
  • The classification applies without further restrictions on the shape of the graphs beyond membership in the family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The work supplies an explicit test for membership in the realizable set once a candidate graph is given.
  • The same flexible family may serve as a starting point for studying degree graphs of groups that are not solvable.

Load-bearing premise

The added flexibility in the construction is sufficient to capture every possible prime character degree graph arising from a solvable group without missing cases or introducing extraneous ones.

What would settle it

A solvable group whose prime character degree graph cannot be obtained from any instance of the flexible construction.

Figures

Figures reproduced from arXiv: 2606.05331 by G. Sivanesan, Jacob Laubacher, Lorenzo Ravaglia, Mark W. Bissler, Thatcher Debowski, Theodore F. Hoelker.

Figure 1
Figure 1. Figure 1: The graphs ΣmL 1,1 for 2 ≤ m ≤ 6 Proof. First note that for m = 2, the graph Σ2L 1,1 is exactly the bowtie graph with five vertices that Lewis constructs in [18], verifying it does indeed occur as the prime character degree graph of some finite solvable group. Next, let m ∈ N such that 3 ≤ m ≤ 35. We follow an approach similar to that done in [18] (see also [3], [14], and [22]). For each appropriate m, we … view at source ↗
Figure 2
Figure 2. Figure 2: The construction of the group Gm Example 3.3. Here we present the explicit construction of the finite solvable group from Proposition 3.2 corresponding to m = 7. Following the entry for m = 7 in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graph ∆(G1 × G7) = Σ7L 1,1 3.2. The case of k = 2. Notice that in the case of k = 2, there are only two options for n due to the restriction that 1 ≤ n ≤ k. For ease of argument, we therefore treat the cases of n = 1 (see [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The graphs Σ2L 2,1 , Σ3L 2,1 , and Σ4L 2,1 Lemma 3.5. Suppose that we are under the conditions of Hypothesis 3.4. Further suppose that Σ (t+1)L 2,1 = ∆(G) for some finite solvable group G, where |G| is minimal. Then G does not have any normal nonabelian Sylow p-subgroup for any p ∈ {a1, a2} ∪ {b1, b3} ∪ {c1, c2, . . . , ct+1}. In particular, p is a strongly admissible vertex. Proof. First we consider the v… view at source ↗
Figure 5
Figure 5. Figure 5: The graphs Σ2L 2,2 , Σ3L 2,2 , and Σ4L 2,2 Corollary 3.10. The graph Σ mL 2,2 does not occur as the prime character degree graph of any solvable group for all m ≥ 2. Proof. Let m ∈ N such that m ≥ 2. We proceed by induction on m. Notice that for the base case of m = 2, the graph Σ2L 2,2 was shown not to occur as the prime character degree graph of any solvable group in [22]. In that paper, following the la… view at source ↗
Figure 6
Figure 6. Figure 6: The graphs Σ2L 3,1 , Σ3L 3,1 , Σ2L 4,1 , and Σ3L 4,1 First we aim to show a1 is admissible. We start by considering the graph determined by removing a1 and all incident edges, which can be denoted by Σ(t+1)L v+1,1 [a1]. Since Σ(t+1)L v+1,1 [a1] is a connected subgraph of Γv+t+1,v+2 with the same vertex set, then Σ(t+1)L v+1,1 [a1] does not occur by the addendum of Theorem 2.8. Next we consider losing edges… view at source ↗
Figure 7
Figure 7. Figure 7: below. c1 c2 a1 a2 a3 b1 b4 b2 b5 b3 c1 c2 a1 a2 a3 b1 b4 b2 b5 b3 b6 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The graphs Σ2R 1,1 , Σ3R 1,1 , and Σ4R 1,1 Proposition 4.3. The graph Σ mR 1,1 occurs as the prime character degree graph of some solvable group for all m ≥ 2. Proof. Let m ∈ N such that m ≥ 2. First, we recall that the singleton K1 occurs as the prime character degree graph of some solvable group G1, and so ∆(G1) = K1. Next, notice that the graph Γm+1,1 occurs as the prime character degree graph of some s… view at source ↗
Figure 9
Figure 9. Figure 9: The graphs Σ0 1,1 , Σ0 2,1 , and Σ0 2,2 All of the aforementioned graphs have been studied and classified before, which we recall below. Proposition 5.1. ([30],[18],[3]) The graphs Σ 0 1,1 , Σ 0 2,1 , and Σ 0 2,2 occur as the prime character degree graph of some solvable group. Proof. Observe that the graph Σ0 1,1 = K3 and therefore has already been shown to occur as the prime character degree graph of som… view at source ↗
Figure 10
Figure 10. Figure 10: The graphs Σ0 3,1 , Σ0 3,2 , and Σ0 3,3 Proposition 5.2. ([14]) The graph Σ 0 3,1 does not occur as the prime character degree graph of any solvable group. Proof. As demonstrated in [14], the graph C20 = Σ0 3,1 does not occur as the prime character degree graph of any solvable group. □ We now turn our attention to the graph Σ0 3,2 , which is a graph with eight vertices and therefore makes an appearance in… view at source ↗
Figure 11
Figure 11. Figure 11: The graph Σ0 3,2 Lemma 5.3. Suppose that Σ 0 3,2 = ∆(G) for some finite solvable group G, where |G| is minimal. Then G does not have any normal nonabelian Sylow bi-subgroup for all 1 ≤ i ≤ 5. In particular, bi is a strongly admissible vertex. Proof. First, let us consider removing the vertex b1 and all incident edges. Doing so produces graph Σ0 3,1 referenced in Proposition 5.2, which does not occur. Now … view at source ↗
Figure 12
Figure 12. Figure 12: The graphs Σ0 4,1 and Σ0 5,1 Lemma 5.9. The graph Σ 0 k,1 does not occur as the prime character degree graph of any solvable group for all k ≥ 4. Proof. Let k ∈ N such that k ≥ 4. We aim to show that the graph Σ0 k,1 does not occur as the prime character degree graph of any solvable group. We will do this by showing that every vertex is admissible. First, we consider the vertex a1. Observe that the graph … view at source ↗
Figure 13
Figure 13. Figure 13: for examples). a1 a2 a3 a4 b1 b2 b3 b4 b5 b6 a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 b6 b7 [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
read the original abstract

We conclude the classification work done in the two previous papers of the same name. Here we add flexibility to the construction, thereby viewing the graphs in full generality. Our goal, as ever, is to determine which graphs do or do not occur as the prime character degree graph of a solvable group.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper concludes the classification work from the two previous papers in the series on prime character degree graphs occurring within a family of graphs. It adds flexibility to the prior construction in order to view the graphs in full generality, with the goal of determining which graphs do or do not occur as the prime character degree graph of a solvable group.

Significance. If the added flexibility indeed yields a complete and gap-free classification without extraneous cases, the result would provide a definitive characterization of prime character degree graphs for solvable groups, advancing the understanding of constraints on character degrees in finite solvable groups.

major comments (1)
  1. [Abstract] Abstract: The central claim that the added flexibility achieves full generality and concludes the classification rests on an unverified assumption that the extended construction captures every possible prime character degree graph arising from a solvable group without omissions or extraneous inclusions; no details, definitions, or arguments supporting this sufficiency are provided in the available text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their comments on the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the added flexibility achieves full generality and concludes the classification rests on an unverified assumption that the extended construction captures every possible prime character degree graph arising from a solvable group without omissions or extraneous inclusions; no details, definitions, or arguments supporting this sufficiency are provided in the available text.

    Authors: The full manuscript provides the required details. Building directly on the constructions and results of the two preceding papers in the series, Section 2 introduces the extended family of graphs with the added flexibility parameters, Section 3 defines the corresponding solvable groups via explicit semidirect products, and Sections 4–5 prove that every prime character degree graph arising from a solvable group is realized by one of these groups (or is excluded by the degree-graph constraints established earlier). These arguments establish both completeness and the absence of extraneous cases; the abstract merely summarizes the resulting classification theorem. revision: no

Circularity Check

0 steps flagged

No significant circularity; classification extends prior construction without reduction to inputs

full rationale

The paper is the third installment in a series and explicitly states it concludes prior classification work by adding flexibility to an existing construction. No equations, fitted parameters, predictions, or self-referential definitions are present in the abstract or stated goal. The central claim is a mathematical classification of which graphs occur as prime character degree graphs of solvable groups; this is achieved by extending a prior construction rather than by any step that reduces by construction to the target result itself. Self-citation to the two previous papers is present but does not serve as the sole justification for a uniqueness theorem or load-bearing premise that would force the outcome. The derivation chain is therefore self-contained as a sequence of group-theoretic constructions and verifications.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no specific free parameters, axioms, or invented entities are identifiable. The work relies on standard results in finite group theory and character theory from prior literature.

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Reference graph

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