Optimal eigenvalues on a metric graph with densities

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 ‐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.