Laplacians on bipartite metric graphs

TL;DR

This paper investigates the spectral properties of Laplacians on finite bipartite metric graphs, revealing eigenvalue coincidences and inequalities that depend on the graph's bipartite structure.

Contribution

It establishes a precise relationship between standard and anti-standard Laplacian eigenvalues on bipartite graphs and explores inequalities involving Dirichlet eigenvalues.

Findings

Positive eigenvalues of standard and anti-standard Laplacians coincide on bipartite graphs

Spectral inequalities for trees are derived from bipartiteness

Connections between eigenvalues and graph bipartiteness are elucidated

Abstract

We study spectral properties of the standard (also called Kirchhoff) Laplacian and the anti-standard (or anti-Kirchhoff) Laplacian on a finite, compact metric graph. We show that the positive eigenvalues of these two operators coincide whenever the graph is bipartite; this leads to a precise relation between their eigenvalues enumerated with multiplicities and including the possible eigenvalue zero. Several spectral inequalities for, e.g., trees are among the consequences of this. In the second part we study inequalities between standard and Dirichlet eigenvalues in more detail and expose another connection to bipartiteness.