An inequality for the heat kernel on an Abelian Cayley graph

TL;DR

This paper establishes a new inequality relating the heat kernel on finite Abelian Cayley graphs to Gaussian functions, solving an open problem and providing insights into heat distribution properties on such graphs.

Contribution

The paper introduces a novel inequality for the heat kernel on Abelian Cayley graphs, connecting it to Gaussian functions and addressing an open problem.

Findings

Proves a new inequality for the heat kernel when t ≤ t'

Links heat kernel behavior to Gaussian functions on lattices

Solves an open problem posed by Regev and Shinkar

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.