Three observations on the Colin de Verdi\`ere spectral graph parameter

TL;DR

This paper discusses properties of the Colin de Verdière spectral graph parameter, providing negative answers to some questions, adding new cases where certain properties hold, and testing conjectures with computational counterexamples.

Contribution

It offers new insights into the properties of CdV matrices, answers a specific open question negatively, and verifies the failure of a conjectured upper bound through computational examples.

Findings

Negative answer to the Perron--Frobenius eigenvector question.

Identification of an additional case where transversality holds.

Counterexamples showing the failure of the conjectured upper bound on μ(G).

Abstract

In this small note, we collect several observations pertaining to the famous spectral graph parameter $\mu$ introduced in 1990 by Y. Colin de Verdi\`ere. 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 $\mu$ by van der Holst, Lov\'asz, 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 Verdi\`ere conjectured an upper bound on $\mu(G)$ 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 $\mu(G)$ for graphs embeddable into 10-torus as well as to several other larger surfaces.