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Three observations on the Colin de Verdière spectral graph parameter

Vojtěch Kaluža, Vadym Koval

Abstract

In this small note, we collect several observations pertaining to the famous spectral graph parameter $μ$ introduced in 1990 by Y. Colin de Verdière. This parameter is defined as the maximum corank among certain matrices akin to weighted Laplacians; we call them CdV matrices. First, we answer negatively a question mentioned in passing in the influential 1996 survey on $μ$ by van der Holst, Lovász, and Schrijver concerning the Perron--Frobenious eigenvector of CdV matrices. Second, by definition, CdV matrices posses certain transversality property. In some cases, this property is known to be satisfied automatically. We add one such case to the list. Third, Y. Colin de Verdière conjectured an upper bound on $μ(G)$ for graphs embeddable into a fixed closed surface. Following a recent computer-verified counterexample to a continuous version of the conjecture by Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault [arXiv:2312.03504], we also check using computer that the analogous example shows the failure of the conjectured upper bound on $μ(G)$ for graphs embeddable into 10-torus as well as to several other larger surfaces.

Three observations on the Colin de Verdière spectral graph parameter

Abstract

In this small note, we collect several observations pertaining to the famous spectral graph parameter introduced in 1990 by Y. Colin de Verdière. This parameter is defined as the maximum corank among certain matrices akin to weighted Laplacians; we call them CdV matrices. First, we answer negatively a question mentioned in passing in the influential 1996 survey on by van der Holst, Lovász, and Schrijver concerning the Perron--Frobenious eigenvector of CdV matrices. Second, by definition, CdV matrices posses certain transversality property. In some cases, this property is known to be satisfied automatically. We add one such case to the list. Third, Y. Colin de Verdière conjectured an upper bound on for graphs embeddable into a fixed closed surface. Following a recent computer-verified counterexample to a continuous version of the conjecture by Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault [arXiv:2312.03504], we also check using computer that the analogous example shows the failure of the conjectured upper bound on for graphs embeddable into 10-torus as well as to several other larger surfaces.

Paper Structure

This paper contains 13 sections, 6 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs.
  4. Vectors and matrices.
  5. Discrete Schrödinger operators.
  6. The Perron--Frobenius eigenvector of CdV matrices
  7. The Strong Arnold Property and connected graphs
  8. Graphs on surfaces
  9. Surfaces, graph embeddings.
  10. Automorphisms, rotary maps.
  11. Discussion and open problems
  12. Strong Arnold Property.
  13. The parameter $\mu$ and graphs on surfaces.

Key Result

Proposition 4

Let $G=(V,E)$ be a connected graph and $M\in\mathcal{M}(G)$ a matrix of corank $2$. Then $M$ satisfies SAP.

Theorems & Definitions (15)

  • Definition 1: Strong Arnold Property (SAP)
  • Proposition 4
  • Conjecture 5: Colin de Verdière YCdV_survey_on_discrete_Schroedinger
  • Conjecture 6: CdV_multiplicity_bounds_mu, CdV_Spectre_Graphes
  • Theorem 7: CdV_multiplicity_bounds_mu
  • Example 8
  • Lemma 9
  • Lemma 10
  • proof
  • proof : Proof of Proposition \ref{['prop:corank_2_SAP']}
  • ...and 5 more