On the Colin de Verdière graph number and penny graphs

Abstract

The Colin de Verdière number of graph G, denoted by $\mu \left(G\right)$, is a spectral invariant of G that is related to some of its topological properties. For example, $\mu \left(G\right)\le 3$ iff G is planar. A penny graph is the contact graph of equal-radii disks with disjoint interiors in the plane. In this note, we prove lower bounds on $\mu \left(G\right)$ when the complement $\bar{G}$ is a penny graph.