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arXiv · 2606.27075

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

Amar Bašić·Lejla Smajlović·Zenan Šabanac
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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.

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Amar Bašić, Lejla Smajlović, Zenan Šabanac. 2026-06-25. Discrete Space-Time Wave Kernels and Trace Identities on Regular Graphs. https://arxiv.org/abs/2606.27075

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