The present book deals with the spectral geometry of infinite graphs. This topic involves the
interplay of three different subjects: geometry, the spectral theory of Laplacians and the heat …

For a given subcritical discrete Schrödinger operator H on a weighted infinite graph X, we
construct a Hardy-weight w which is optimal in the following sense. The operator H− λ w is …

We study solutions of the generalized porous medium equation on infinite graphs. For
nonnegative or nonpositive integrable data, we prove the existence and uniqueness of mild …

A few years ago various disparities for Laplacians on graphs and manifolds were
discovered. The corresponding results are mostly related to volume growth in the context of …

Criticality theory on graphs Page 1 J. Spectr. Theory 10 (2020), 73–114 DOI 10.4171/JST/286
Journal of Spectral Theory © European Mathematical Society Criticality theory for Schrödinger …

We analyze properties of semigroups generated by Schrödinger operators Δ− V or
polyharmonic operators−(− Δ) m, on metric graphs both on L p-spaces and spaces of …

The main focus in this memoir is on Laplacians on both weighted graphs and weighted
metric graphs. Let us emphasize that we consider infinite locally finite graphs and do not …

In this expository chapter we answer two fundamental questions concerning discrete
magnetic Schrödinger operator associated with weighted graphs. We discuss when formal …

The goal of this article is to survey various results concerning stochastic completeness of
graphs. In particular, we present a variety of formulations of stochastic completeness and …

We study the p-independence of spectra of Laplace operators on graphs arising from
regular Dirichlet forms on discrete spaces. Here, a sufficient criterion is given solely by a …