Constructing solvable groups whose character degree graphs generalize the bowtie

Abstract: We present here a generalized construction of a finite solvable group whose prime character degree graph has the shape and structure of the bowtie graph. As with the original bowtie, the graphs obtained by this generalized construction, under certain restrictions, cannot be realized by the usual method of taking direct products of smaller graphs. Within the condition of $n=1$, we show how this recovers the original bowtie graph, which has five vertices. We also provide examples and explicit choices of primes which generate graphs with more vertices.

• The paper constructs solvable groups with prime character degree graphs that generalize the bowtie, enabled by replacing single vertices with complete subgraphs of primes determined via specific factorizations.
• The authors use Frobenius actions and Gallagher's theorem to determine the character degrees ensuring specific block structures.
• They verify a range of determining parameters from small values computed explicitly to larger values abstractly expressed, and identify an upper level where construction falls apart.

Overview

This paper by Laubacher, Lewis, Ravaglia, and Summers addresses a recurring problem in the character theory of finite solvable groups: exhibiting solvable groups whose prime character degree graphs realize specific combinatorial shapes that cannot be obtained by direct-product constructions. The authors generalize the construction, due to Lewis in her classification of five-vertex graphs (2608.19374), of a solvable group whose prime character degree graph $\Delta(G)$ is the "bowtie" graph. The generalized construction replaces each of three single vertices with clusters (complete subgraphs) of primes determined by factorizations of $2^{\hat{p}}-1$ and $(2^{\hat{p}}+1)/3$, producing graphs with $n+m+k+2$ vertices. The resulting graphs are deliberately constrained so that they cannot be decomposed as direct products of smaller occurring graphs.

The paper situates itself within a substantial body of classification work: for graphs on up to eight vertices, occurring graphs are rare. The authors tabulate the aggregate state of knowledge — for example, among 12,346 graphs on eight vertices, only 39 occur, 12,103 do not occur, and 204 remain unclassified. This scarcity motivates constructions of solvable groups realizing particular shapes, following the precedent of Lewis's diameter-three construction (2608.19374) and Dugan's generalization thereof, which enabled the classification of diameter-three graphs on seven and eight vertices.

The construction

Fix $n \in \mathbb{N}$ and let $\hat{p} = p_1 p_2 \cdots p_n$ be a product of distinct primes, none equal to 2 or 3, subject to the condition that

$$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$

factor into distinct primes, all different from 2, 3, and the primes dividing $\hat{p}$. The key identity underlying the construction is

$$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$

The group $G$ is built from a quotient of a skew polynomial ring over $2^{\hat{p}}-1$0 by the ideal generated by $2^{\hat{p}}-1$1. Specifically, with $2^{\hat{p}}-1$2 and $2^{\hat{p}}-1$3 the image of $2^{\hat{p}}-1$4, one sets $2^{\hat{p}}-1$5, a group of order $2^{\hat{p}}-1$6 whose derived subgroup is $2^{\hat{p}}-1$7. Letting $2^{\hat{p}}-1$8 be the cyclic multiplicative group of $2^{\hat{p}}-1$9 (of order $(2^{\hat{p}}+1)/3$0) and $(2^{\hat{p}}+1)/3$1 the Galois group of $(2^{\hat{p}}+1)/3$2 over its base field (cyclic of order $(2^{\hat{p}}+1)/3$3), the group is defined as $(2^{\hat{p}}+1)/3$4, where $(2^{\hat{p}}+1)/3$5 has order $(2^{\hat{p}}+1)/3$6 and satisfies $(2^{\hat{p}}+1)/3$7.

The computation of $(2^{\hat{p}}+1)/3$8 proceeds via standard techniques: orbit-stabilizer analysis on $(2^{\hat{p}}+1)/3$9 yields degrees divisible by $n+m+k+2$0, while the Frobenius action of $n+m+k+2$1 on $n+m+k+2$2, together with full ramification of nonprincipal characters of $n+m+k+2$3 over $n+m+k+2$4 (established via Problem 6.12 of Isaacs' text), Gallagher's theorem, and an analysis of fixed fields under subgroups of the Galois group, yields degrees involving $n+m+k+2$5. The final result is

$n+m+k+2$6

Since all primes involved are distinct, the corresponding prime character degree graph has $n+m+k+2$7 vertices, organized as two complete blocks ($n+m+k+2$8 and $n+m+k+2$9) joined to a central complete block $n \in \mathbb{N}$0. When the clusters are condensed to single vertices, this is precisely a bowtie shape: two triangles sharing the central vertex.

Non-redundancy via Pálfy's inequality

A notable feature of the paper is its careful accounting for when the construction produces genuinely new graphs versus graphs realizable by direct products. The Zsigmondy theorem imposes constraints on the factor counts: $n \in \mathbb{N}$1, and $n \in \mathbb{N}$2.

The upper bound on $n \in \mathbb{N}$3 is tied directly to Pálfy's inequality for disconnected graphs, which states that if a disconnected $n \in \mathbb{N}$4 has components of sizes $n \in \mathbb{N}$5, then $n \in \mathbb{N}$6. If $n \in \mathbb{N}$7, then the component containing vertex 3 would have size $n \in \mathbb{N}$8 where $n \in \mathbb{N}$9, so Pálfy's inequality permits such a graph — and indeed the authors show it is nothing more than a direct product of $\hat{p} = p_1 p_2 \cdots p_n$0 with a disconnected complete-bipartite-component graph. Demanding $\hat{p} = p_1 p_2 \cdots p_n$1 therefore ensures the constructed graph falls outside direct-product realizability, which is the entire point of the construction.

Analogously, when $\hat{p} = p_1 p_2 \cdots p_n$2 exceeds the minimum, extra $\hat{p} = p_1 p_2 \cdots p_n$3-vertices can always be appended via a direct product with the singleton graph $\hat{p} = p_1 p_2 \cdots p_n$4, so taking the minimal $\hat{p} = p_1 p_2 \cdots p_n$5 is advantageous. The authors illustrate both phenomena concretely:

Case	Parameters	Realization
$\hat{p} = p_1 p_2 \cdots p_n$6, $\hat{p} = p_1 p_2 \cdots p_n$7, $\hat{p} = p_1 p_2 \cdots p_n$8	$\hat{p} = p_1 p_2 \cdots p_n$9: $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$0, $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$1	Original bowtie (5 vertices)
$$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$2, $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$3, $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$4	$$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$5: $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$6	Direct product $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$7
$$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$8, $$\hat{q} = 2^{\hat{p}} - 1 = q_1 q_2 \cdots q_m \quad \text{and} \quad \hat{r} = \frac{2^{\hat{p}}+1}{3} = r_1 r_2 \cdots r_k$$9, $\hat{p}$0	$\hat{p}$1: Mersenne $\hat{p}$2, $\hat{p}$3, $\hat{p}$4	Direct product $\hat{p}$5
$\hat{p}$6, $\hat{p}$7, $\hat{p}$8	e.g., $\hat{p}$9, $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$0, $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$1	Genuinely new graphs

For $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$2 exceeding the bound, the resulting six-vertex graph equals $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$3 where $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$4 is the disconnected graph with complete components of sizes two and three. For $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$5 with minimal $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$6, the three admissible values $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$7 yield explicit examples with fully verified prime factorizations, including very large primes such as $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$8 in the case $$2^{2\hat{p}} - 1 = (2^{\hat{p}}-1)(2^{\hat{p}}+1) = 3\,\hat{q}\,\hat{r}.$$9, $G$0.

The $G$1, $G$2 case recovers Lewis's original bowtie exactly: taking $G$3 reproduces her choice of primes, and other admissible values ($G$4) yield additional five-vertex bowties.

Limitations and open questions

The paper is candid about several restrictions. First, the construction depends on the number-theoretic condition that both $G$5 and $G$6 split into sufficiently many distinct primes avoiding 2 and 3; the authors note that some sets of primes fail this condition and pose as an open question whether there are infinitely many sets satisfying it. Second, for $G$7, achieving the minimum $G$8 may be impossible: citing Conjecture 4.3 of Dolfi–Hafezieh–Spiga, the authors observe that one expects $G$9 in that range. In their own computations for $2^{\hat{p}}-1$00, the smallest $2^{\hat{p}}-1$01 found was 11 (with $2^{\hat{p}}-1$02, yielding $2^{\hat{p}}-1$03), well above both $2^{\hat{p}}-1$04 and the conjectured minimum of 8; additional examples give $2^{\hat{p}}-1$05 and $2^{\hat{p}}-1$06. Whether values closer to the conjectured minimum are attainable remains unresolved. Third, as $2^{\hat{p}}-1$07 grows, the sheer density of edges makes the full graphs unwieldy, which is why the paper adopts the condensed notation with $2^{\hat{p}}-1$08, $2^{\hat{p}}-1$09, $2^{\hat{p}}-1$10 standing for clusters rather than individual vertices.

Conclusion

The paper provides a parametric family of solvable groups whose prime character degree graphs generalize the bowtie graph from five to arbitrarily many vertices, with explicit prime choices verifying feasibility at every level of the parameter $2^{\hat{p}}-1$11 presented. Its main technical contribution is the verification, through a combination of Frobenius actions, full ramification arguments, and Gallagher's theorem, that the resulting degree sets produce exactly the intended block structure. Equally important is the negative bookkeeping: by invoking Pálfy's inequality and the Zsigmondy bounds, the authors delineate precisely which outputs of the construction are irreducible to direct products, thereby identifying the genuinely new occurring graphs. The construction extends the toolbox available for classifying occurring prime character degree graphs by vertex count, complementing the direct-product method that accounts for most known occurrences but cannot reach these graphs.