Related papers: On torsional rigidity and ground-state energy of c…
We investigate the torsion function or landscape function and its integral, the torsional rigidity, of Laplacians on metric graphs subject to $\delta$-vertex conditions. A variational characterization of torsional rigidity and Hadamard-type…
We study torsional rigidity for graph and quantum graph analogs of well-known pairs of isospectral non-isometric planar domains. We prove that such isospectral pairs are distinguished by torsional rigidity.
We study the $p$-\emph{torsion function} and the corresponding $p$-\emph{torsional rigidity} associated with $p$-Laplacians and, more generally, $p$-Schr\"odinger operators, for $1<p<\infty$, on possibly infinite combinatorial graphs. We…
We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for…
We prove a quantitative uncertainty principle at low energies for the Laplacian on fairly general weighted graphs with a uniform explicit control of the constants in terms of geometric quantities. A major step consists in establishing lower…
A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian in Gaussian space is derived for sets with prescribed generalized torsional rigidity. The result provides an extension of the classical spectral inequality due to…
The objective of this note is to provide an interpretation of the discrete version of Morse inequalities, following Witten's approach via supersymmetric quantum mechanics, adapted to finite graphs, as a particular instance of Morse-Witten…
We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of…
The gauge approach to gravity based on the local Lorentz group with a general independent affine connection A_{\mu cd} is developed. We consider SO(1,3) gauge theory with a Lagrangian quadratic in curvature as a simple model of quantum…
We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which…
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a…
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). We prove a closed formula of combinatorial flavour, as it is expressed as a sum…
In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null…
Motivated by the connection between the first eigenvalue of the Dirichlet-Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional…
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace formulae is obtained which link together two…
Formal Laplace operators are analyzed for a large class of resistance networks with vertex weights. The graphs are completed with respect to the minimal resistance path metric. Compactness and a novel connectivity hypothesis for the…
Measurement incompatibility--the impossibility of jointly measuring certain quantum observables--is a fundamental resource for quantum information processing. We develop a graph-theoretic framework for quantifying this resource for large…
We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph.…
We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how various types of changes of a local or…
We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums…