Optimal eigenvalues on a metric graph with densities

TL;DR

This paper studies Laplacians on finite metric graphs with measures, establishing existence of minimal eigenvalues, their geometric characterization, and connections to spectral optimization, unifying various graph operator frameworks.

Contribution

It introduces a framework for spectral optimization on metric graphs with densities, proving existence of minimal eigenvalues and linking them to graph geometry.

Findings

Existence of minimal k-th eigenvalues for the first eigenvalue

Geometric characterization of the first optimal eigenvalue via resistance metric

Weyl law for higher optimal eigenvalues

Abstract

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 operators: discrete graph Laplacians, Kirchhoff Laplacians and Dirichlet-to-Neumann operators on graphs. Our main interest lies in spectral optimization with respect to the underlying measure. In contrast to the setting of domains and manifolds, we prove that a minimal $k$-th eigenvalue exists, whereas the corresponding maximization problem has no meaning. We then establish connections between these optimal eigenvalues and the geometry of the metric graph, including a transparent geometric characterization of the first optimal eigenvalue via the resistance metric, and a Weyl law for the higher optimal eigenvalues.