Faber-Krahn inequality for the heat content on quantum graphs via random walk expansion

We study the heat content on quantum graphs and investigate whether an analogon of the Rayleigh-Faber-Krahn inequality holds. This means that heat content at time $T$ among graphs of equal volume would be maximized by intervals (the graph analogon of balls as in the classic Rayleigh-Faber-Krahn inequality). We prove that this holds at extremal times, that is at small and at large times. For this, we employ two complementary approaches: In the large time regime, we rely on a spectral-theoretic approach, using Mercer's theorem whereas the small-time regime is dealt with by a random walk approach using the Feynman-Kac formula and Brownian motions on metric graphs. In particular, in proving the latter, we develop a new expression for the heat content as a positive linear combination of expected return times of (discrete) random walks - a formulation which seems to yield additional insights compared to previously available methods such as the celebrated Roth formula and which is crucial for our proof. The question whether a Rayleigh-Faber-Krahn inequality for the heat content on metric graphs holds at all times remains open.