On the eigenvalue estimates for the weighted Laplacian on metric graphs

TL;DR

This paper investigates eigenvalue estimates for the weighted Laplacian on metric graphs, establishing inequalities that depend only on the graph's length and total weight, regardless of the graph's shape.

Contribution

It provides a novel eigenvalue inequality for weighted Laplacians on metric graphs that is independent of the graph's topology.

Findings

Eigenvalues satisfy an inequality involving total length and weight

Results are applicable to various graph structures

Generalizations extend the inequality's scope

Abstract

Eigenvalue behavior for the equation -\lambda y"=Vu on the edges of a graph G of final total length, with a non-negative weight function V and under the Kirchhoff matching conditions at the vertices and zero boundary condition at at least one point of G, is studied. It is shown that the eigenvalues satisfy an inequality which involves the length |G| and the total mass corresponding to V but otherwise does not depend on the graph. Applications and generalizations of this result are also discussed.