On extremal eigenvalues of the graph Laplacian

TL;DR

This paper characterizes graphs with maximally degenerate Laplacian eigenvalues, called lasso trees, and compares these to existing eigenvalue estimates, revealing structural conditions for degeneracy.

Contribution

It introduces the class of lasso trees and provides a characterization of graphs with maximally degenerate eigenvalues, linking eigenvalue multiplicity to graph structure.

Findings

Lasso trees are the graphs with maximally degenerate eigenvalues.

Maximal eigenvalue multiplicity is achieved only by lasso trees.

The paper relates eigenvalue estimates to graph degeneracy conditions.

Abstract

Upper and lower estimates of eigenvalues of the Laplacian on a metric graph have been established in 2017 by G. Berkolaiko, J.B. Kennedy, P. Kurasov and D. Mugnolo. Both these estimates can be achieved at the same time only by highly degenerate eigenvalues which we call maximally degenerate. By comparison with the maximal eigenvalue multiplicity proved by I. Kac and V. Pivovarchik in 2011 we characterize the family of graphs exhibiting maximally degenerate eigenvalues which we call lasso trees, namely graphs constructed from trees by attaching lasso graphs to some of the vertices.