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 …

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on
weighted graphs. We first show that, for locally finite graphs and a certain family of metrics …

We consider the quantum completeness problem, ie the problem of confining quantum
particles, on a non-complete Riemannian manifold M equipped with a smooth measure …

We present and prove new characterizations of parabolicity and stochastic completeness for
a general weighted manifold M as well as the uniqueness of the Markov extensions of the …

We study the evolution of the heat and of a free quantum particle (described by the
Schrödinger equation) on two-dimensional manifolds endowed with the degenerate …

We investigate the relationship between one of the classical notions of boundaries for
infinite graphs, graph ends, and self‐adjoint extensions of the minimal Kirchhoff Laplacian …

Motivated by the recent development in the theory of jump processes, we investigate its
conservation property. We will show that a jump process is conservative under certain …

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 paper, we introduce the notion of oriented faces especially triangles in a connected
oriented locally finite graph. This framework then permits to define the Laplace operator on …

A Liouville property and its application to the Laplacian of an infinite graph Page 117
Contemporary Mathematics Volume 484 , 2009 A Liouville property and its application to the …