Constructing heat kernels on infinite graphs

Let G be an infinite, edge- and vertex-weighted graph with certain reasonable restrictions. We construct the heat kernel of the associated Laplacian using an adaptation of the parametrix approach due to Minakshisundaram-Pleijel in the setting of Riemannian geometry. This is partly motivated by the wish to relate the heat kernels of a graph and a subgraph, or of a domain and a discretization of it. As an application, assuming that the graph is locally finite, we express the heat kernel HG (x, y;t) as a Taylor series with the lead term being a (x, y) tʳ, where r is the combinatorial distance between x and y and a (x, y) depends (explicitly) upon edge and vertex weights. In the case G is the regular (q+1) -tree with q 1, our construction reproves different explicit formulas due to Chung-Yau and to Chinta-Jorgenson-Karlsson. Assuming uniform boundedness of the combinatorial vertex degree, we show that a dilated Gaussian depending on any distance metric on G, which is uniformly bounded from below can be taken as a parametrix in our construction. Our work extends in part the recent articles LNY21, CJKS23 in that the graphs are infinite and weighted.