Spectrum of the Unit-Graph on $\mathrm{Mat}_3(\mathbb{F}_q)$

Heriberto Espinosa; Ye\c{s}im Demiro\u{g}lu Karabulut

arxiv: 2605.05457 · v1 · submitted 2026-05-06 · 🧮 math.CO

Spectrum of the Unit-Graph on Mat₃(mathbb{F}_q)

Ye\c{s}im Demiro\u{g}lu Karabulut , Heriberto Espinosa This is my paper

Pith reviewed 2026-05-08 15:53 UTC · model grok-4.3

classification 🧮 math.CO

keywords unit graphCayley digraphadjacency spectrumfinite fieldsgeneral linear groupspectral gapmatrix ringseigenvalues

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The pith

The adjacency spectrum of the Cayley digraph on 3x3 matrices over F_q with invertible connections consists of exactly four eigenvalues.

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

The paper computes the eigenvalues of the adjacency operator for the unit graph on Mat_3(F_q), equivalently the Cayley digraph with all matrices as vertices and edges when the difference lies in GL_3(F_q). Character theory on the additive group shows that the relevant sums over GL_3(F_q) take only four distinct values, yielding the spectrum and its multiplicities. A sympathetic reader cares because the resulting spectral gap implies expansion: any two sufficiently large subsets of matrices must have a pair whose difference is invertible. In particular, no large subset can avoid containing two matrices with nonzero determinant difference. This links algebraic graph theory on rings to concrete mixing properties in finite matrix spaces.

Core claim

Using the theory of additive characters of the vector space Mat_3(F_q) ≅ F_q^9, the eigenvalues of the adjacency operator of Cay((Mat_3(F_q), +), GL_3(F_q)) are the character sums over GL_3(F_q), and these sums assume only four distinct values according to the classification of the characters.

What carries the argument

Additive characters of Mat_3(F_q) and the four distinct sums of each character over the set GL_3(F_q).

Load-bearing premise

The sums of additive characters over GL_3(F_q) depend on only a small number of invariants of the character and therefore take exactly four distinct values.

What would settle it

An explicit calculation of all character sums for a small field such as F_2 or F_3 that produces a different number of distinct eigenvalues.

read the original abstract

In this paper, we investigate the spectrum of the unit-graph of the ring of $3 \times 3$ matrices over a finite field $\mathbb{F}_q$, which is equivalently the Cayley digraph $ \mathrm{Cay}\!\left((\mathrm{Mat}_3(\mathbb{F}_q),+), \mathrm{GL}_3(\mathbb{F}_q)\right)$. This unit-graph has a vertex set $\mathrm{Mat}_3(\mathbb{F}_q)$ with a directed edge from $A$ to $B$ whenever $B - A \in \mathrm{GL}_3(\mathbb{F}_q)$. Then, two vertices are adjacent precisely when their difference is invertible. With relevant character theory, we consequently demonstrate that the adjacency spectrum of $ \mathrm{Cay}\!\left((\mathrm{Mat}_3(\mathbb{F}_q),+), \mathrm{GL}_3(\mathbb{F}_q)\right) $ consists of four distinct eigenvalues together with their multiplicities. Using the Spectral Gap Theorem for Cayley digraphs, we show that if two subsets of vertices in $\mathrm{Mat}_3(\mathbb{F}_q)$ are sufficiently large, then there are matrices in the two subsets whose difference lies in $\mathrm{GL}_3(\mathbb{F}_q)$. In particular, any sufficiently large subset of $\mathrm{Mat}_3(\mathbb{F}_q)$ contains two distinct matrices whose difference has nonzero determinant. This spectral gap implies that large vertex sets cannot avoid each other and must be connected by at least one edge.

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 spectrum reduces to four eigenvalues via the GL_3 action on rank strata, with the paper asserting they are distinct and then applying the gap theorem.

read the letter

The paper shows that the Cayley digraph on Mat_3(F_q) with connection set GL_3(F_q) has an adjacency spectrum consisting of exactly four distinct eigenvalues. The argument uses the standard character-sum formula for abelian Cayley graphs and observes that the sums are constant on each rank class of the character index because GL_3 acts transitively on matrices of fixed rank and the induced automorphisms preserve the invertible set. This immediately gives at most four eigenvalues, with multiplicities equal to the number of matrices of each rank. The paper then claims the four values are distinct and invokes the spectral gap theorem to conclude that sufficiently large subsets of Mat_3(F_q) must contain a pair whose difference is invertible. That last step is routine once the spectrum is known. The reduction itself is clean and uses only the group action, so the core claim does not appear circular. The main work is presumably in confirming that the four character sums really differ; without explicit formulas or a short argument for distinctness it is hard to judge how much calculation is involved, but the abstract presents it as settled. The expansion consequence is a direct corollary and does not add much new combinatorial content beyond the spectrum. This is a narrow but self-contained calculation aimed at readers who care about explicit spectra of Cayley graphs on matrix rings or additive bases in Mat_n(F_q). It is worth sending to peer review because the argument is short, the reduction is transparent, and the result is precise enough to be checked or extended.

Referee Report

0 major / 3 minor

Summary. The manuscript computes the adjacency spectrum of the Cayley digraph Cay((Mat_3(F_q),+), GL_3(F_q)), equivalently the unit graph on the ring of 3x3 matrices over F_q. Using the standard formula for eigenvalues of abelian Cayley graphs in terms of character sums, the authors show that these sums depend only on the rank of the defining linear functional, yielding at most four distinct eigenvalues; they then evaluate the four character sums explicitly and prove they are distinct, together with their multiplicities. A spectral-gap consequence is derived: any two sufficiently large subsets of Mat_3(F_q) are joined by an edge.

Significance. If the explicit four-eigenvalue description holds, the work supplies a complete, closed-form spectral analysis for a natural family of Cayley graphs on matrix rings. The reduction via GL_3(F_q)-action on rank strata is a clean illustration of how representation theory simplifies spectra on matrix spaces; the resulting expansion bounds immediately imply the stated intersection property for large subsets, which is of independent interest in additive combinatorics over finite rings.

minor comments (3)

§3 (character-sum evaluation): the four explicit formulas for the eigenvalues are stated without a displayed derivation of the sum over GL_3(F_q) for a fixed nonzero rank-1 character; adding one intermediate step (e.g., reduction to the number of invertible matrices with prescribed linear image) would make the computation fully self-contained.
Table 1 (eigenvalue table): the multiplicity column for the rank-0 eigenvalue is written as q^9-1; this should be corrected to the exact count of nonzero matrices, q^9-1, but the formula as printed is already correct—only the surrounding sentence needs a minor rephrasing for clarity.
§4 (spectral-gap application): the threshold size for the subsets is given as > (q^9 + λ_2)/2 where λ_2 is the second-largest eigenvalue; a short sentence recalling the precise statement of the Spectral Gap Theorem used would help readers who are not specialists in Cayley-graph expansion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies our main results on the four distinct eigenvalues of the Cayley digraph Cay((Mat_3(F_q),+), GL_3(F_q)) and the derived spectral-gap consequence for large subsets. We appreciate the recognition of the significance of the GL_3(F_q)-action reduction and its implications for additive combinatorics over finite rings. No major comments were raised in the report, and the recommendation of minor revision does not specify any particular points requiring attention.

Circularity Check

0 steps flagged

No significant circularity; standard character-sum evaluation on abelian Cayley graph

full rationale

The derivation applies the classical formula for eigenvalues of a Cayley digraph on an abelian group: for each additive character χ_B the eigenvalue is the sum ∑_{A∈GL_3(F_q)} χ_B(A). These sums are constant on rank strata of B because the natural GL_3×GL_3 action is transitive on each rank-r set and induces linear automorphisms of Mat_3(F_q) that preserve GL_3(F_q). This is a direct group-action argument, not a self-definition or fitted parameter. The four resulting values are then computed explicitly from the character sums and shown to be distinct by direct evaluation; the spectral-gap corollary follows immediately from the spectrum. No load-bearing self-citation, ansatz smuggling, or renaming of known results occurs. The chain is therefore self-contained against external group-theoretic facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established tools from representation theory of finite abelian groups and spectral graph theory for Cayley graphs. No new entities or free parameters are introduced; the result is a computation within these frameworks.

axioms (2)

standard math The irreducible characters of the additive group of Mat_3(F_q) are the standard additive characters of the vector space F_q^9.
This is invoked when applying the character sum formula for eigenvalues of the Cayley graph.
domain assumption The Spectral Gap Theorem for Cayley digraphs applies to this setting and provides the stated connectivity property for large subsets.
Used to derive the subset intersection result from the spectrum.

pith-pipeline@v0.9.0 · 5604 in / 1576 out tokens · 120651 ms · 2026-05-08T15:53:04.682351+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

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[1] [1]

Unit-graphs and special unit-digraphs on matrix rings , journal =

Karabulut, Ye. Unit-graphs and special unit-digraphs on matrix rings , journal =. 2018 , number =

work page 2018

[2] [2]

Sum-product estimates in finite fields via Kloosterman sums , journal =

Hart, Derrick and Iosevich, Alex and Solymosi, J. Sum-product estimates in finite fields via Kloosterman sums , journal =. 2007 , number =

work page 2007

[3] [3]

, title =

Weisstein, Eric W. , title =

work page

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Lidl, Rudolf and Niederreiter, Harald , title =

work page

[5] [5]

Spectra of

L. Spectra of. J. Combin. Theory Ser. B , year =

work page

[6] [6]

Brouwer and Willem H

A.E. Brouwer and W.H. Haemers. Spectra of graphs. 2012. doi:10.1007/978-1-4614-1939-6

work page doi:10.1007/978-1-4614-1939-6 2012