Optimal eigenvalues on a metric graph with densities
Kiyan Naderi, Noema Nicolussi
TL;DR
Problem: optimize the $k$-th eigenvalue $\lambda_k(H_\mu)$ of Laplacians on a finite metric graph $G$ over probability measures $\mu$. Approach: develop a unified framework with $H_\mu$, establish the existence of minimizers, and derive geometric and spectral characterizations of the optimizers; show continuity under weak-* convergence and connect to graph resistance geometry. Key findings: the first optimum satisfies $\lambda_1^{\min}(G)=4/\operatorname{diam}_r(G)$ with two-point minimizers, and a Weyl law $\lambda_k^{\min}(G)\sim 4k^2/L(G)$ for large $k$; the path graph yields explicit spectra $\lambda_k^{\min}(P)=4k^2$ and explicit minimizers, while Dirichlet-to-Neumann and spectral-partitioning perspectives illuminate the structure. Significance: unifies discrete, Kirchhoff, and DtN operators within a spectral-optimization framework on graphs, tying extremal eigenvalues to resistance geometry and partitioning theory, with precise asymptotics and exact solutions in key cases.
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.