On the Colin de Verdiere graph number and penny graphs

TL;DR

This paper investigates the Colin de Verdiere number, a spectral graph invariant, focusing on lower bounds when the graph's complement is a penny graph, linking spectral properties to geometric contact graphs.

Contribution

It establishes new lower bounds on the Colin de Verdiere number for graphs whose complements are penny graphs, connecting spectral invariants with geometric graph representations.

Findings

Lower bounds on (G) when (G) complement is a penny graph

Links between spectral invariants and geometric contact graphs

Insights into topological properties via spectral bounds

Abstract

The Colin de Verdiere number of graph G, denoted by \mu(G), is a spectral invariant of G that is related to some of its topological properties. For example, \mu(G) \leq 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(G) when the complement \bar{G} is a penny graph.