An inequality for the heat kernel on an Abelian Cayley graph

Abstract

We demonstrate a relationship between the heat kernel on a finite weighted Abelian Cayley graph and Gaussian functions on lattices. This can be used to prove a new inequality for the heat kernel on such a graph: when $t \leq t'$, \[ \frac{H_t(u, v)} {H_t(u,u)} \leq \frac{H_{t'}(u, v)} {H_{t'}(u,u)}. \] This was an open problem posed by Regev and Shinkar.