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Generalized Paley graphs on finite fields have real spectra precisely when they are undirected.

2026-05-10 18:22 UTC

load-bearing objection The paper cleanly shows generalized Paley graphs have real spectrum exactly when undirected and that only oriented Paley digraphs have three eigenvalues, with a cyclotomic method yielding new infinite integral families.

arxiv 2604.06513 v1 submitted 2026-04-07 math.CO

The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields

Ricardo A. Podest\'a , Denis E. Videla This is my paper
classification math.CO
keywords generalized Paley graphsspectrumfinite fieldsCayley graphseigenvaluesWaring numbersdirected graphsintegrality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper examines generalized Paley graphs Γ(k,q) on the finite field of order q, formed as Cayley graphs whose connection set is exactly the subgroup of k-th powers. It proves that the spectrum consists entirely of real numbers if and only if the graph is undirected, which holds precisely when the connection set is closed under inversion. The work also classifies the directed cases, showing that they always possess at least three distinct eigenvalues and that the oriented Paley graphs together with their disjoint unions are the only ones having exactly three. As an application the graphs are used to reduce the computation of weak Waring numbers over finite fields to the ordinary Waring numbers.

Core claim

We characterize all GP-graphs having real spectrum; namely, Spec(Γ(k,q)) ⊂ ℝ if and only if Γ(k,q) is undirected. We study conditions for integrality and give a method to produce integral GP-graphs through cyclotomic polynomials, yielding several infinite families. For directed GP-digraphs we show they always have three or more eigenvalues and prove that the only ones with exactly three are the oriented Paley graphs or disjoint unions of them. We further show that generically these digraphs have period 1, except for Γ(q-1,q) with q odd, which decomposes into oriented p-cycles of period p. As an application we reduce the computation of weak Waring numbers over finite fields to the computation

What carries the argument

The Cayley graph on the additive group of F_q whose connection set is the subgroup of k-th powers, whose eigenvalues are obtained from the standard character-sum formula over that subgroup.

Load-bearing premise

The spectrum of each generalized Paley graph is completely determined by the standard character-sum formula for abelian Cayley graphs with no further restrictions imposed by the prime power q.

What would settle it

A single directed generalized Paley graph whose spectrum is entirely real, or a weak Waring number over a finite field whose value differs from the corresponding classic Waring number.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Infinite families of integral generalized Paley graphs arise from cyclotomic polynomials.
  • Directed generalized Paley graphs always possess at least three distinct eigenvalues.
  • The oriented Paley graphs and their disjoint unions are the unique directed generalized Paley graphs with exactly three eigenvalues.
  • All directed generalized Paley graphs except Γ(q-1,q) for odd q have index of imprimitivity one.
  • Weak Waring numbers over finite fields reduce directly to ordinary Waring numbers via the graph construction.

Where Pith is reading between the lines

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

  • The real-spectrum criterion may extend to other families of Cayley graphs on finite fields whose connection sets are subgroups.
  • The reduction of weak Waring numbers could simplify explicit calculations for small exponents or small fields.
  • The period-one result for most directed cases suggests these digraphs are strongly connected or have simple orbit structures under the field automorphism group.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies generalized Paley graphs Γ(k,q) = Cay(F_q, (F_q^*)^k) for q = p^m. It proves that the spectrum is real if and only if the graph is undirected (equivalently, -1 lies in the subgroup of k-th powers). It gives a cyclotomic-polynomial construction that produces infinite families of integral GP-graphs. For the directed case it shows that every GP-digraph has at least three distinct eigenvalues and that exactly three occur only for oriented Paley graphs or disjoint unions of them. It determines the periods of these digraphs and, as an application, reduces the computation of weak Waring numbers over finite fields to that of classical Waring numbers.

Significance. The spectral classification rests on standard character-sum formulas for abelian Cayley graphs and is therefore immediately verifiable. The explicit construction of integral examples and the reproof of the Waring-number reduction via graph spectra are useful contributions that link spectral graph theory with additive combinatorics over finite fields. The fact that the central claims follow from well-known facts on normal adjacency matrices and multiplicative subgroups strengthens the manuscript.

minor comments (4)
  1. The definition of the period (or index of imprimitivity) of a digraph is used in §5 but is not recalled in the text; a one-sentence reminder with the standard definition would help readers.
  2. The statement that GP-digraphs generically have period 1 would benefit from an explicit small example (e.g., q=13, k=3) showing the eigenvalue multiplicities and the resulting period.
  3. The cyclotomic-polynomial construction in §3 is presented for prime-power q; it would be useful to state explicitly whether the same polynomials work when q is not prime.
  4. A reference to the 2012 Cochrane-Cipra paper is given for the Waring-number result, but the manuscript does not indicate which parts of their proof are reproduced or simplified by the graph-theoretic approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The significance of the spectral classification, the construction of integral examples, and the reduction of weak Waring numbers via graph spectra is appreciated. We note the recommendation for minor revision and will prepare an updated version incorporating any necessary clarifications or corrections.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivations rest on the standard character-sum formula for the eigenvalues of abelian Cayley graphs (via the Fourier basis of additive characters of F_q) and basic facts about when such an adjacency matrix is Hermitian. The claim that the spectrum is real if and only if the graph is undirected follows immediately from A being normal and A = A^* precisely when the connection set S satisfies S = -S, i.e., -1 lies in the subgroup of k-th powers; this is a direct group-theoretic equivalence with no fitted parameters or self-referential definitions. The further statements that directed GP-digraphs have at least three eigenvalues and exactly three only for oriented Paley graphs (or their disjoint unions) are obtained by counting the distinct values taken by the relevant multiplicative character sums over subgroups of different indices; these counts rely on established properties of cyclotomic cosets and Gauss sums in finite fields, not on any reduction to the paper's own inputs. The application to weak Waring numbers is explicitly noted as reproducing a prior independent result of Cochrane and Cipra. No self-citations are load-bearing for the central claims, and no quantities are defined in terms of themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established facts from finite-field arithmetic and representation theory of abelian groups without introducing new free parameters or postulated entities.

axioms (2)
  • standard math The multiplicative group of a finite field is cyclic.
    Used to define the subgroup of k-th powers and its index.
  • standard math Eigenvalues of a Cayley graph on an abelian group are given by the character sums over the connection set.
    Fundamental tool for computing the spectrum in §2 and §3.

pith-pipeline@v0.9.0 · 5623 in / 1334 out tokens · 50364 ms · 2026-05-10T18:22:15.002223+00:00 · methodology

0 comments
read the original abstract

We consider the family of generalized Paley graphs (GP-graphs for short) $\Gamma(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(\Gamma(k,q)) \subset \mathbb{R}$ if and only if $\Gamma(k,q)$ is undirected. We then study conditions for integrality in the spectrum and give a general method to produce integral GP-graphs through cyclotomic polynomials. Using this, we construct several infinite families of integral GP-graphs. Next, we focus on directed GP-graphs (GP-digraphs). We show that GP-digraphs always have three or more eigenvalues, and then we prove that there is only one kind of GP-digraphs having three different eigenvalues: the oriented Paley graphs $\vec{\mathcal{P}}_q$ or disjoint unions of copies of them, $\vec{\mathcal{P}}_q \cup \cdots \cup \vec{\mathcal{P}}_q$. Then, we show that generically the GP-digraphs have period 1 (equivalently index of imprimitivity 1) except for $\Gamma(q-1,q)$ with $q$ odd, which is the disjoint union of oriented $p$-cycles, having period $p$. Finally, as an application, we study weak Waring numbers over finite fields through GP-graphs. In particular, we reduce the computation of the weak Waring numbers over finite fields to the computation of classic Waring numbers over finite fields, a result previously obtained by Cochrane and Cipra in 2012 by other means.

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