On the heat content of compact quantum graphs

TL;DR

This paper derives a comprehensive formula for the heat content on compact metric graphs with boundary conditions, revealing geometric insights and establishing comparison principles for different graph topologies.

Contribution

It introduces a novel combinatorial formula for heat content on metric graphs and develops new surgery and comparison principles.

Findings

Closed combinatorial formula for heat content

Small-time asymptotic expansion with geometric information

Comparison principles for different graph topologies

Abstract

We study the heat content for Laplacians on compact, finite metric graphs with Dirichlet conditions imposed at the "boundary" (i.e., a given set of vertices). We prove a closed formula of combinatorial flavour, as it is expressed as a sum over all closed orbits hitting the boundary. Our approach delivers a small-time asymptotic expansion that delivers information on crucial geometric quantities of the metric graph, much in the spirit of the celebrated corresponding result for manifolds due to Gilkey-van den Berg; but unlike other known formulae based on different methods, ours holds for all times $t>0$ and it displays stronger decay rate in the short time limit. Furthermore, we prove new surgery principles for the heat content and use them to derive comparison principles for the heat content between metric graphs of different topology.