Abstract
We study heat kernels of locally finite graphs and discrete heat equation morphisms. These are combinatorial analogs to heat equation morphisms in Riemannian geometry (cf. E. Loubeau, [10]), parallel closely the discrete harmonic morphisms due to H. Urakawa, [13], and their properties are related to the initial value problem for the discrete heat equation. In applications we consider Hamming graphs (using the discrete Fourier calculus on Z2N), establish a heat kernel comparison theorem, and study the maps of ε-nets induced by heat equation morphisms among two complete Riemannian manifolds.
