Discrete diffusion-type equation on regular graphs and its applications

• Published in: Journal of difference equations and applications Vol. 29; no. 4; pp. 455 - 488
• Main Authors: Cadavid, Carlos A., Hoyos, Paulina, Jorgenson, Jay, Smajlović, Lejla, Vélez, Juan D.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 03-04-2023; Taylor & Francis Ltd
• Subjects: Apexes; Bessel function; Bessel functions; Formulas (mathematics); Graphs; Heat kernel; Probability distribution; Random walk; regular tree and graph; trace formula
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236198.2023.2219784

We derive an explicit formula for the fundamental solution
to the discrete-time diffusion equation on the
-regular tree
in terms of the discrete I-Bessel function. We then use the formula to derive an explicit expression for the fundamental solution
to the discrete-time diffusion equation on any
-regular graph X. Going further, we develop three applications. The first one is to derive a general trace formula that relates the spectral data on X to its topological data. Though we emphasize the results in the case when X is finite, our method also applies when X has a countably infinite number of vertices. As a second application, we obtain a closed-form expression for the return time probability distribution of the uniform random walk on any
-regular graph. The expression is obtained by relating
to the uniform random walk on a
-regular graph. We then show that if
is a sequence of
-regular graphs whose number of vertices goes to infinity and which satisfies a certain natural geometric condition, then the limit of the return time probability distributions from
is equal to the return time probability distribution on the tree
. As a third application, we derive formulas which express the number of distinct closed irreducible walks without tails on a finite graph X in terms of moments of the spectrum of its adjacency matrix.