Bounds on the negative eigenvalues of Laplacians on finite metric graphs

TL;DR

This paper derives bounds on the negative eigenvalues of Laplacians on finite metric graphs, providing inequalities for compact cases and general bounds for non-compact graphs, advancing spectral analysis in graph theory.

Contribution

It introduces new bounds on negative eigenvalues of Laplacians on finite metric graphs, including Poincaré-type inequalities for compact graphs.

Findings

Derived bounds on negative eigenvalues for finite metric graphs.

Established Poincaré-type inequalities for compact finite metric graphs.

Extended spectral analysis tools for graph Laplacians.

Abstract

For a self--adjoint Laplace operator on a finite, not necessarily compact, metric graph lower and upper bounds on each of the negative eigenvalues are derived. For compact finite metric graphs Poincar\'{e} type inequalities are given.