The heat kernel on the diagonal for a compact metric graph

TL;DR

This paper derives explicit formulas for the heat kernel on compact metric graphs, analyzes its small-time behavior, and introduces an edge heat trace formula linking eigenfunctions to graph loops.

Contribution

It provides a new explicit formula for the heat kernel on metric graphs and establishes an edge heat trace formula based on loop sums, advancing spectral analysis techniques.

Findings

Explicit heat kernel formulas for compact metric graphs

Edge heat trace formula relating eigenfunctions and loops

Explicit solutions for symmetric graphs

Abstract

We analyze the heat kernel associated to the Laplacian on a compact metric graph, with standard Kirchoff-Neumann vertex conditions. An explicit formula for the heat kernel as a sum over loops, developed by Roth and Kostrykin, Potthoff, and Schrader, allows for a straightforward analysis of small-time asymptotics. We show that the restriction of the heat kernel to the diagonal satisfies a modified version of the heat equation. This observation leads to an "edge" heat trace formula, expressing the a sum over eigenfunction amplitudes on a single edge as a sum over closed loops containing that edge. The proof of this formula relies on a modified heat equation satisfied by the diagonal restriction of the heat kernel. Further study of this equation leads to explicit formulas for completely symmetric graphs.