arxiv: 2603.15851 · v2 · submitted 2026-03-16 · 🧮 math.GR · math.CO

Recognition: no theorem link

Classifying Prime Character Degree Graphs With Eight Vertices

Mark L. Lewis , Andrew Summers

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Pith reviewed 2026-05-15 10:11 UTC · model grok-4.3

classification 🧮 math.GR math.CO

keywords prime character degree graphsfinite solvable groupsgraph classificationcharacter degreesconnected graphsdirect productsdiameter three

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The pith

Finite solvable groups realize only 37 of the 11,117 connected eight-vertex prime character degree graphs.

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

The paper classifies which prime character degree graphs with eight vertices can arise from finite solvable groups. It combines prior results into a general algorithm that determines realizability, showing 37 connected graphs occur while 56 do not. Of the realizable cases, 34 come from direct products and three have diameter three. This matters because the graphs encode the prime factors of character degrees and thus constrain possible group structures. The work fully classifies all disconnected graphs but leaves 206 connected graphs undecided.

Core claim

Among the 11,117 non-isomorphic connected graphs with eight vertices, exactly 37 occur as the prime character degree graph of some finite solvable group, with 34 constructed via direct products and three having diameter three, while 56 graphs are shown not to occur for any finite solvable group.

What carries the argument

The general algorithm built from compiled known constructions and prohibitions, which systematically checks whether each eight-vertex graph can be realized as a prime character degree graph of a finite solvable group.

If this is right

Direct products of smaller solvable groups account for nearly all realizable eight-vertex graphs.
Graphs of diameter three can occur, extending beyond the diameter-two cases studied earlier.
Certain families of graphs are prohibited by existing theorems and remain impossible.
All 1,229 disconnected eight-vertex graphs are classified as possible or impossible.
The same algorithmic approach applies in principle to graphs of any order.

Where Pith is reading between the lines

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

Applying the algorithm to nine or more vertices would reveal whether the proportion of realizable graphs changes with size.
The undecided graphs may require new non-direct-product constructions to decide.
The classification supplies concrete test cases for broader questions about bounding the number of distinct prime divisors in character degrees of solvable groups.

Load-bearing premise

The known results and algorithm together account for every possible way a solvable group could realize one of these graphs without missing additional constructions.

What would settle it

Finding even one finite solvable group whose prime character degree graph is one of the 206 undecided graphs would show that more than 37 graphs occur.

Figures

Figures reproduced from arXiv: 2603.15851 by Andrew Summers, Mark L. Lewis.

**Figure 2.** Figure 2: The two disconnected graphs which do not occur [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: The two disconnected graphs which occur 4.1 Component Sizes 7 & 1 Consider the finite field of order 2491 acted on by its full multiplication group followed by its Galois group. That is, take G1 = F2 491 ⋊ F× 2 491 ⋊ Gal(F2 491 /F2). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: The occurring disconnected graph with component sizes seven and one [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: The occurring disconnected graph with component sizes six and two [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: Graphs Ai for 1 ≤ i ≤ 7, all of which occur 5.2 Graphs of Clique Sizes Six & Two There are forty-five graphs covered by cliques of sizes six and two. These graphs are listed in appendix B. In this section, we show that nineteen of these graphs occur; sixteen of which will be constructed as direct products, while the remaining three follow the construction in [7] for appropriate choices of primes. Additiona… view at source ↗

**Figure 7.** Figure 7: The graphs Bi for i ∈ I Note that graph B2 is the direct product of the disconnected graph with component sizes five and two with K1. Meanwhile, B37 is the direct product of the disconnected graph with component sizes five and one with the empty (no edges) graph of order two. Graph B38 can also be realized as the direct product of two disconnected graphs, namely the disconnected graph with component sizes … view at source ↗

**Figure 8.** Figure 8: Graphs B7, B13, and B36 which occur Let Φn(x) be the n th cyclotomic polynomial and choose distinct primes p, q, and r such that q < r and Φq(p), Φr(p), and Φqr(p) are pairwise coprime. We note here that for the constructions of B13 and B36, it is sufficient to choose q = 3, which results in a simpler computation of character degrees. For graph B7, however, a choice of primes which produced the desired gra… view at source ↗

**Figure 9.** Figure 9: The graphs ∆(G13) = B13 and ∆(G36) = B36 As mentioned previously, computing the character degrees when q > 3 is more complicated, as there are more characters whose kernels correspond to the hyperplanes discussed in [7]. Under these circumstances, we have the following set of character degrees: 15 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗

**Figure 10.** Figure 10: The graph ∆(G7) = B7 16 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗

**Figure 11.** Figure 11: The diameter three graphs B1, B3, B30, and B34 which do not occur We first note that for each graph, there is exactly one pair of vertices which are distance three apart. Following [14] and [24], we label these p1 and p4. Recall that ρ3 is the set of vertices which are distance two from p1. In theorem two of [24], Sass showed that an occurring diameter three graph must satisfy |ρ3|≥ 3. For graphs B1 and B… view at source ↗

**Figure 12.** Figure 12: Graph B4 = Γ6,2, which does not occur We note that the graph B4 has two vertices of order two which are adjacent to one another and share no common neighbor. Thus, B4 can also be eliminated by the main result of [4] and therefore falls into two previously studied families. Unclassified Graphs Of the forty-five graphs covered by cliques of sizes six and two, twenty-one remain unclassified. These are the gr… view at source ↗

**Figure 13.** Figure 13: The unclassified graphs Bj for j ∈ J As was the case in [11], one major source of difficulty in dealing with the unclassified graphs above is the fact that they contain the occurring disconnected graph with component sizes six and two as a subgraph. Additionally, several graphs have connected subgraphs of order seven which either occur or remain unclassified based off of the work in [11]. 5.3 Graphs of Cl… view at source ↗

**Figure 14.** Figure 14: The graphs Ci for i ∈ I Note that the graph C35 is the direct product of the disconnected graph with component sizes four and two with the empty graph of order 2. Meanwhile graph C61 is the direct product of the disconnected graph with component sizes three and two with the disconnected graph having component sizes two and one. Graph C112 is the product of the bowtie graph (shown to occur by Lewis in [18]… view at source ↗

**Figure 15.** Figure 15: The graphs Cj for j ∈ J1 which fail to satisfy |ρ3|≥ 3 We note as well that the graph C1 has two cut vertices and may also be eliminated by the main result of [16]. Denote the remaining seventeen graphs by Cj for j ∈ J2 := {13, 16, 18, 19, 25, 29, 31, 32, 38, 39, 43, 49, 51, 53, 70, 86, 88}. Noting that each of these graphs satisfies |ρ1 ∪ ρ2|≥ 3, it follows from theorem 4 of [24] that |ρ3 ∪ ρ4|≥ 2 3 = 8 … view at source ↗

**Figure 16.** Figure 16: The graphs Cj for j ∈ J2 Previously Classified Graphs There are two graphs covered by cliques of sizes five and three that have been previously classified. Specifically, these are graphs C14 and C50 which can be seen in appendix C and figure 17 below [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗

**Figure 17.** Figure 17: The graphs C14 and C50 which were previously classified Notice that graph C14 is isomorphic to the graph Γ5,3 which was shown not occur in [2]. Additionally, graph C50 is isomorphic to the graph ΣR 3,1 of [6], which does not occur by the main result of that paper. Unclassified Graphs One-hundred sixteen graphs covered by cliques of sizes five and three currently require additional study, and possibly new … view at source ↗

**Figure 18.** Figure 18: Graph D96, which can be constructed via direct products Diameter Three Graphs When considering graphs covered by cliques of sizes four and four, there are twentyone which have diameter three. Namely the graphs Dj where j ∈ {1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 18, 20, 22, 28, 36, 37, 40, 53, 55}, which can be found in figures 19 and 20 as well as appendix D. As was the case when studying graphs covere… view at source ↗

**Figure 19.** Figure 19: The graphs Dj for j ∈ J1 which do not occur Again, we point out that graph D1 can also be eliminated via the main result of [16] as it has two cut vertices. Next, we examine the graphs Dj where j ∈ J2 := {3, 5, 8, 9, 11, 13, 14, 16, 18, 22, 28, 36, 37, 40, 53, 55}. For each of these graphs, we have that |ρ1 ∪ ρ2|≥ 3 which implies |ρ3 ∪ ρ4|≥ 8. Thus, by the same argument used in the previous section, we ha… view at source ↗

**Figure 20.** Figure 20: The graphs Dj for j ∈ J2 which do not occur Previously Classified Graphs There are four graphs covered by cliques of sizes four and four which have been previously studied. Namely, the graphs D15, D43, D77, and D86 of figure 22. All four graphs also appear in appendix D [PITH_FULL_IMAGE:figures/full_fig_p025_20.png] view at source ↗

**Figure 22.** Figure 22: The previously studied graphs D15, D43, D77 and D86 Notice that D15 is isomorphic to the graph ΣL 3,1 which was shown not occur in [10]. Also, the graph D43 is simply the graph Γ4,4 which was eliminated by the main result of [2]. The graphs D77 and D86 are both 5-regular which cannot occur by theorem A of [21]. Note that this makes the complete graph K8 and the graph D96 (which is 6-regular) the only regu… view at source ↗

**Figure 23.** Figure 23: The non-occurring graph D19 with all admissible vertices Lemma 5.1. The graph D19 of figure 22 has all admissible vertices and therefore does not occur. Proof. For convenience, we label the vertices as in figure 22, with vertices in the same clique having indices of the same parity. For any choice of pi and pj in the same clique (except possibly p1p7), there exists pk in the opposite clique such that pipk… view at source ↗

read the original 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.

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.

Desk Editor's Note private letter to a colleague

The paper delivers a partial but systematic classification of eight-vertex prime character degree graphs for solvable groups, realizing 37 connected examples and ruling out 56 while leaving 206 open.

read the letter

The main takeaway is that this work turns a collection of known constructions and prohibitions into one algorithm and runs it on all 12,346 graphs of order eight. It fully settles the 1,229 disconnected cases, produces explicit groups for 37 connected graphs (mostly direct products, three with diameter three), and shows 56 graphs cannot occur. That is the concrete advance: the first organized list for this size rather than scattered examples in the literature. The algorithm itself is the useful piece, since the authors frame it as reusable for larger orders. The undecided 206 graphs are presented honestly as open, which keeps the claims proportionate. The reliance on prior results for the constructions and prohibitions is standard in this subfield and the counts line up with what the abstract reports. One limitation is that the paper does not describe running the algorithm on the known complete classifications for orders four through six to confirm it reproduces those results exactly. Without that check, it is harder to be sure the order-eight run has no missed realizations or incorrect prohibitions. The work is aimed at researchers already tracking prime character degree graphs of solvable groups. It does not open new methods or settle a broad open question, but it does organize the small-case data in one place. For that narrow audience the paper is worth referee time so the gaps can be addressed in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript compiles known results from the literature on prime character degree graphs of finite solvable groups to develop a general algorithm for classifying such graphs of any order. It then applies the algorithm to all 12,346 non-isomorphic graphs on eight vertices, fully classifying the 1,229 disconnected graphs, exhibiting explicit constructions for 37 connected graphs that occur (34 via direct products and 3 of diameter three), prohibiting 56 graphs, and leaving the status of 206 graphs undecided.

Significance. If the algorithm is verified to be correct, the work advances the program of classifying prime character degree graphs by providing a reusable general method, explicit constructions for 37 realizable order-8 graphs, and prohibitions for 56 others. These concrete results and the algorithmic framework constitute a useful contribution that can be extended to higher orders.

major comments (1)

[§3] The section describing the algorithm does not include any cross-validation against the known complete classifications for graphs on 4, 5, or 6 vertices. Because the reported counts for order 8 (37 occurring, 56 prohibited) rest entirely on the algorithm's faithful reproduction of all literature constructions and prohibitions, the absence of this check leaves open the possibility that some of the 37 constructions are missed or that some of the 56 prohibitions are erroneous.

minor comments (2)

[Abstract] The abstract and introduction would be clearer if they stated the precise numbers (37 occurring, 56 prohibited, 206 undecided) in the opening paragraph rather than burying them later.
A summary table listing the final counts for disconnected, occurring, prohibited, and undecided graphs would improve readability of the main results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript's contribution and for the constructive major comment. We address it point by point below and will incorporate the requested verification in the revised version.

read point-by-point responses

Referee: [§3] The section describing the algorithm does not include any cross-validation against the known complete classifications for graphs on 4, 5, or 6 vertices. Because the reported counts for order 8 (37 occurring, 56 prohibited) rest entirely on the algorithm's faithful reproduction of all literature constructions and prohibitions, the absence of this check leaves open the possibility that some of the 37 constructions are missed or that some of the 56 prohibitions are erroneous.

Authors: We agree that explicit cross-validation would strengthen the presentation. In the revised manuscript we will add a dedicated subsection to §3 that runs the algorithm on all non-isomorphic graphs of orders 4, 5, and 6 and confirms that every known construction and prohibition from the literature is recovered exactly. This verification will directly support the reliability of the 37 constructions and 56 prohibitions reported for order 8. revision: yes

Circularity Check

0 steps flagged

No circularity: classification rests on compiled external literature and explicit constructions

full rationale

The paper compiles known results and constructions from the literature, develops a general algorithm, and applies it to the 11,117 connected graphs on eight vertices. It reports 37 realizable graphs (34 via direct products, 3 of diameter three) with explicit constructions, 56 prohibited graphs, and 206 undecided. No derivation step reduces by construction to the paper's own inputs, fitted parameters, or self-citation chain; all load-bearing claims are grounded in external literature results and concrete group constructions rather than self-definitional or renamed quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification relies on standard theorems of finite solvable group theory and graph theory already present in the literature; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Standard results on character degrees of solvable groups and prime graphs from the cited literature
The algorithm is built by compiling these prior theorems.

pith-pipeline@v0.9.0 · 5424 in / 1180 out tokens · 41223 ms · 2026-05-15T10:11:28.940424+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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