Planarity criteria for metric graphs

TL;DR

This paper explores the Colin de Verdière parameter for metric graphs, showing it can be characterized by the maximal multiplicity of the second eigenvalue of Laplacians with delta couplings, extending previous discrete graph results.

Contribution

It introduces a new metric graph framework to determine the Colin de Verdière parameter via eigenvalue multiplicities, linking discrete and metric graph spectral properties.

Findings

The Colin de Verdière parameter equals the maximal second eigenvalue multiplicity for certain Laplacians on metric graphs.

Multiple families of Laplacians and Schrödinger operators are shown to yield the same Colin de Verdière number.

The approach bridges discrete graph invariants and continuous metric graph spectral theory.

Abstract

The Colin de Verdi\`ere parameter is a number assigned to discrete graphs which equals the maximal multiplicity of the second eigenvalue of a certain family of Laplacian matrices related to the graph. In this paper it is shown that the Colin de Verdi\`ere parameter can be obtained in the setting of metric graphs by looking at the maximal multiplicity of the second eigenvalue for Laplacians on metric graphs with delta couplings at the vertices. Two different families of Laplacians, as well as a family of Schr\"odinger operators, all leading to the Colin de Verdi\`ere number are presented.