Discrete diffusion-type equation on regular graphs and its applications

Abstract

ABSTRACT 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.