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 $\mu$ 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.