The Parametrix Construction of the Heat Kernel on a Graph

In this paper we develop the parametrix approach for constructing the heat kernel on a graph G. In particular, we highlight two specific cases. First, we consider the case when G is embedded in a Euclidean domain \(\Omega \) $\mathrm{\Omega }$ , and we use a heat kernel associated to \(\Omega \) $\mathrm{\Omega }$ to obtain a formula for the heat kernel on G. Second, we consider when G is a possibly infinite subgraph of a larger graph \(\widetilde{G}\) $\tilde{G}$ , and we obtain a formula for the heat kernel on G from the heat kernel on \(\widetilde{G}\) $\tilde{G}$ restricted to G.