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arxiv: 2606.27075 · v1 · pith:MB7HR3K3new · submitted 2026-06-25 · 🧮 math.CO · math.CA· math.SP

Discrete Space-Time Wave Kernels and Trace Identities on Regular Graphs

Amar Ba\v{s}i\'c , Lejla Smajlovi\'c , Zenan \v{S}abanac This is my paper

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classification 🧮 math.CO math.CAmath.SP
keywords discrete wave equationregular graphswave kernelsmodified Bessel functionsnon-backtracking walkstrace formulascombinatorial identitiesChebyshev polynomials
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The pith

Wave kernels on regular graphs are expressed using discrete modified Bessel functions and non-backtracking walk counts, enabling new trace formulas.

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

The paper derives explicit formulas for the fundamental solutions of the discrete space-time wave equation on regular graphs using the forward time-difference scheme. These solutions, called wave kernels, are written in terms of discrete modified Bessel functions and counts of non-backtracking walks on the graph. This creates a direct connection between wave propagation and the graph's combinatorial structure. The uniqueness of the wave kernel is then used to establish a new trace-type formula for the affine Laplace-type operator. This formula produces various combinatorial identities, including closed forms for certain twisted trigonometric sums and Chebyshev polynomial sums.

Core claim

For the forward time-difference scheme on a (q+1)-regular graph X, the two fundamental solutions of the discrete space-time wave equation associated with the affine Laplace-type operator are given explicitly by formulas involving discrete modified Bessel functions and the non-backtracking walk counts on X. This link allows the proof of a new trace-type formula using the uniqueness property of the wave kernel, which in turn yields combinatorial identities such as closed-form expressions for trigonometric sums twisted by an additive character and finite sums of Chebyshev polynomials twisted by binomial coefficients.

What carries the argument

The wave kernels for the forward time-difference scheme, defined via discrete modified Bessel functions and non-backtracking walk counts on the regular graph.

Load-bearing premise

The wave kernel satisfies a uniqueness property that permits deriving the trace-type formula from it.

What would settle it

Direct computation of the wave solution on a small regular graph such as the complete graph K_{q+2} and checking whether it matches the proposed formula involving Bessel functions and walk counts.

read the original abstract

We study the discrete space-time wave equation on a $(q+1)$-regular graph $X$ associated with the affine Laplace-type operator. For the forward time-difference scheme we derive explicit formulas for the two fundamental solutions (wave kernels) in terms of discrete modified Bessel functions and the non-backtracking walk counts on $X$ thus providing a direct and explicit link between wave propagation and combinatorial graph data. Utilizing uniqueness property of the wave kernel, we prove a new trace-type formula associated to the affine Laplace-type operator on $X$ and apply it to deduce many combinatorial identities. For example, we derive a closed-form expression for evaluation of some trigonometric sums twisted by an additive character as well as evaluations of finite sums of Chebyshev polynomials twisted by binomial coefficients.

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.

Referee Report

0 major / 4 minor

Summary. The paper derives explicit formulas for the two fundamental solutions (wave kernels) of the forward time-difference scheme for the discrete space-time wave equation on a (q+1)-regular graph X, expressed via discrete modified Bessel functions and non-backtracking walk counts. It then uses the uniqueness property of the wave kernel to establish a new trace-type formula for the affine Laplace-type operator and applies this to obtain combinatorial identities, including closed forms for trigonometric sums twisted by additive characters and finite sums of Chebyshev polynomials twisted by binomial coefficients.

Significance. If the derivations are correct, the work establishes a direct combinatorial interpretation of discrete wave propagation on regular graphs and yields new trace identities that generate explicit evaluations of twisted sums. This strengthens the link between spectral graph theory, non-backtracking walks, and discrete analysis, with potential applications to expander graphs and combinatorial enumeration. The explicit, parameter-free character of the kernel formulas (when they hold) and the deduction of identities from uniqueness are notable strengths.

minor comments (4)
  1. §2, Definition of the affine Laplace-type operator: clarify the precise normalization and the role of the parameter q in the adjacency operator to ensure the wave equation is unambiguously stated before the kernel derivations.
  2. §4, Theorem on the trace formula: the uniqueness argument for the wave kernel should include an explicit reference to the initial conditions or boundary behavior used to pin down the solution uniquely.
  3. Figure 1 and the accompanying discussion of non-backtracking walks: the caption should state the graph size and regularity explicitly so that the combinatorial counts are reproducible from the figure alone.
  4. The application section deriving the twisted Chebyshev sums: add a short remark on how the trace identity specializes when the additive character is trivial, recovering a known identity for verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation of minor revision. We are pleased that the combinatorial and spectral connections are viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives explicit wave-kernel formulas from the forward time-difference scheme using discrete modified Bessel functions and non-backtracking walk counts, then invokes the uniqueness property of the wave kernel to obtain a trace identity. No quoted equations or steps reduce a claimed prediction or result to a fitted parameter, self-definition, or self-citation chain by construction. The combinatorial link is presented as a direct construction from the discrete scheme, and the trace formula follows from an external uniqueness property rather than an internal redefinition. The derivation chain remains self-contained against the stated combinatorial inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the central derivation rests on the uniqueness property of the wave kernel.

axioms (1)
  • domain assumption Uniqueness property of the wave kernel for the discrete space-time wave equation on regular graphs
    Invoked to deduce the trace-type formula from the explicit kernels.

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