A mathematical preprint reports that the spectral gap of a class of rotation graphs fades asymptotically for complete bipartite graphs and complete split graphs. In plain terms, the separation between the leading part of the spectrum and the next part becomes smaller as these graph families grow. The work studies rotation graphs R(G) built from connected graphs G.
A broad lower bound
The study considers connected graphs G with n + 1 vertices, for n at least 2, and examines the second-largest adjacency eigenvalue of their rotation graph R(G). An adjacency eigenvalue is a number that captures part of a graph’s connectivity pattern through its adjacency matrix. Across the stated class, the paper establishes the lower bound λ2 ≥ n − 2. That is a guaranteed floor, not an exact description of the second-largest eigenvalue for every graph.
To obtain the bound, the authors use the Rayleigh-quotient characterization of λ2, a way of identifying eigenvalue bounds by comparing how a matrix acts on carefully chosen vectors. They also use Cheeger’s inequalities, which relate edge expansion, a measure of how readily a graph can be divided by cutting edges, to the second eigenvalue in a regular graph. These tools connect two views of the same structure, one algebraic and one based on cuts.
Adding twin vertices changes expansion
The analysis examines what happens in a true-twin construction, denoted Gtv. For the resulting graph, the edge expansion of its rotation graph obeys h(R(Gtv)) ≤ 2|R(G)|/|R(Gtv)|. The expression is an upper bound: it gives a ceiling on the expansion after the construction, rather than an exact value.
A corresponding false-twin construction gives a strict bound, h(R(Gfv)) < 2|R(G)|/(|R(Gfv)| − |R(G)|), for the stated construction with graphs of size at least 2. Together, the two results give explicit limits on the edge expansion of the resulting rotation graphs.
A closer look at the stellohedron
The paper then turns to the stellohedron, written S_n = R(K1,n), and uses an equitable-partition quotient to make its spectrum easier to study. This quotient compresses structurally similar parts of the graph into a smaller matrix, while its eigenvalues are retained in the spectrum of the full graph. The transformed quotient matrix produces a Sturm sequence of principal-minor polynomials, allowing the researchers to locate eigenvalues within specified intervals.
For n greater than 1, the first five eigenvalues of the quotient, listed from largest to smaller, are placed in successive one-unit intervals: the i-th lies in (n − i, n − i + 1] for i from 1 through 5. Those quotient values are also eigenvalues of the full stellohedron, but their ranks within the quotient should not be read as proof that they are the first five eigenvalues of the full graph.
For n at least 2, the stellohedron’s edge expansion is reported to satisfy h(R(K1,n)) < 2/(n − 1), while its second-largest eigenvalue satisfies λ2 > n − 4/(n − 1). These are strict bounds, not exact values. The formulas show that the study is tracking expansion and eigenvalues on different scales as n grows.
What remains unresolved
The spectral picture is not complete. The paper reports an eigenvalue μ near n with multiplicity at least n − 1 under the inequality displayed in its theorem, alongside the leading eigenvalue λ1 = n. Its conclusion also identifies two additional small eigenvalues. The exact smallest eigenvalue, however, is left unresolved and is described as computationally suggested rather than proved.
As an exploratory check, a computer calculation at n = 20 evaluated the sign-sequence quantities across the stated range of k and found s(n − i) = i for i from 1 through 15. That calculation is a single reported parameter check, not a general proof.
The work is theoretical, with mathematically defined graphs and adjacency matrices as its units of analysis. Its general λ2 statement is a lower bound rather than an exact value, and its localization of the first five eigenvalues directly concerns the equitable quotient, not necessarily the first five eigenvalues of the full stellohedron. The smallest-eigenvalue result also remains unresolved beyond the reported computational suggestion.
Publication status
The document is an arXiv version 1 preprint dated 25 Aug 2026. It presents theoretical bounds, quotient-spectrum localization and an exploratory computation for a narrowly defined family of mathematical graphs.
Paper data and sources
Original title: On the spectrum and expansion of graph associahedra
Authors: Ana Gargantini, Adrián Pastine, Pablo Torres, Mario Valencia-Pabon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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