In this thesis we present several new results concerning the diffusion and its behaviour on networks in the form of combinatorial and metric graphs. In addition, a selection of new results is presented which allow for the comparison of two Schrödinger operators carrying boundary (i.e., vertex) conditions of the same class with respect to their spectra and thus, we also provide some new contributions to spectral theory on quantum graphs. Beginning with the investigation of the so-called heat kernel on metric graphs and in particular its regularity properties, the heat content, which can be understood as the space integration of the heat kernel corresponding to the heat equation with Dirichlet and Neumann boundary conditions will be studied. Among others, we derive a heat content formula stating that the heat content can be represented as a summation over directed paths on a graph that start and end at the same Dirichlet vertex. This represents the counterpart to the celebrated and well-known path sum formula which is originally due to J. P. Roth from 1983 for the heat kernel itself, which yields that the heat kernel on metric graphs can be represented as a summation of so-called scattering and Gaussian terms over all directed paths between any two points on the graph. By using this heat content formula, novel small-time asymptotics for the heat content on quantum graphs will be discussed, complementing known results on domains and manifolds. A central observation, which is due to the fact that we are working on graphs, is an exponential and thus much faster decay rate of the remainder term for small times than the polynomial decay rate which is known on domains and manifolds. Apart from this, the interplay between the geometric structure of the underlying graph and the heat content and thus the diffusion is investigated in more detail. In particular, we will derive an extremal Faber-Krahn inequality, revealing that the heat content of a graph is maximised at small and large times by a path of same total volume. Subsequently, the so-called torsional rigidity, which is given by the L1 norm of the solution - also known as the torsion function - of a certain Poisson equation of the form -Δu = 1. Alternatively, the torsional rigidity can be understood as the space-time integration of the heat kernel and therefore it also provides further insights into the behaviour of diffusion processes on graphs. Based on already known results on quantum graphs, the torsional rigidity is mainly investigated on combinatorial graphs, also in the non-linear case. This includes the derivation of several surgery and comparison principles, as well as a selection of upper and lower bounds on the torsional rigidity, which are mostly of geometric flavour and additionally lead to novel estimates on the bottom of the p-spectrum - which, within mathematical physics often goes under the name of ground state energy. This is concluded by the proof of a so-called Kohler-Jobin inequality in the linear case, in other words, a uniform estimate on the (scaled) product consisting of the first Dirichlet eigenvalue on the one hand, and the torsional rigidity on the other: it has to be emphasised that - so far - such estimates are only known in the case of Euclidean domains and quantum graphs, therefore combinatorial graphs become a third ambient, where a Kohler-Jobin inequality is known to hold. Finally, several results concerning the spectral comparison of two Schrödinger operators on quantum and combinatorial graphs are presented. These will include the case of infinite graphs, but will also provide some new contributions to the realm of inverse spectral theory in form of an "Ambarzumian-type" result.
Patrizio Bifulco (Thu,) studied this question.
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