On the Heat Content of Compact Quantum Graphs

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) and standard conditions imposed elsewhere. We prove a closed formula of combinatorial flavor, as it is expressed as a sum over all paths starting and ending at boundary vertices. By delivering a small-time asymptotic expansion, our approach yields 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 $$t>0$$ t > 0 and it displays a 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.