A family of diameter-based eigenvalue bounds for quantum graphs

TL;DR

This paper derives sharp lower bounds for the first non-trivial eigenvalue of the Laplacian on quantum graphs based on diameter and length, extending previous results and addressing open problems.

Contribution

It introduces new diameter-based eigenvalue bounds for quantum graphs, including higher eigenvalues, resolving an open problem and complementing existing bounds.

Findings

Established sharp lower bounds for the first eigenvalue.

Extended bounds to higher eigenvalues under certain length conditions.

Results are sharp for trees.

Abstract

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. This extends a result of, and resolves an open problem from, [J. B. Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17 (2016), 2439--2473, Section 7.2], and also complements an analogous lower bound for the corresponding eigenvalue of the combinatorial Laplacian on a discrete graph. We also give a family of corresponding lower bounds for the higher eigenvalues under the assumption that the total length of the graph is sufficiently large compared with its diameter. These inequalities are sharp in the case of trees.