The parametrix construction of the heat kernel on a graph

TL;DR

This paper develops a parametrix method for constructing the heat kernel on graphs, focusing on cases where the graph is embedded in a domain or a larger graph, providing explicit formulas in these scenarios.

Contribution

It introduces a novel parametrix approach for heat kernel construction on graphs, extending existing methods to embedded and subgraph cases.

Findings

Formulas for heat kernels on graphs embedded in Euclidean domains.

Formulas for heat kernels on subgraphs derived from larger graphs.

Extension of parametrix methods to graph settings.

Abstract

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 Eulidean domain or manifold $\Omega$, and we use a heat kernel associated to $\Omega$ to obtain a formula for the heat kernel on $G$. Second, we consider when $G$ is a subgraph of a larger graph $\widetilde{G}$, and we obtain a formula for the heat kernel on $G$ from the heat kernel on $\widetilde{G}$ restricted to $G$.