An inequality for the heat kernel on an Abelian Cayley graph

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 $t\le t^{\mathrm{\prime }}$, $$\frac{H_t(u,v)}{H_t(u,u)}\le \frac{H_{t^{\mathrm{\prime }}}(u,v)}{H_{t^{\mathrm{\prime }}}(u,u)}$$ This was an open problem posed by Regev and Shinkar.