Sphere Representations, Stacked Polytopes, and the Colin de Verdière Number of a Graph

Lon Mitchell,Lynne Yengulalp

doi:10.37236/4444

Abstract

We prove that a $k$-tree can be viewed as a subgraph of a special type of $(k+1)$-tree that corresponds to a stacked polytope and that these "stacked'' $(k+1)$-trees admit representations by orthogonal spheres in $\mathbb{R}^{k+1}$. As a result, we derive lower bounds for Colin de Verdière's $\mu$ of complements of partial $k$-trees and prove that $\mu(G) + \mu(\overline{G}) \geq |G| - 2$ for all chordal $G$.

Highlights

Yves Colin de Verdiere’s graph invariant μ is defined as the maximum nullity over a special class of real symmetric matrices [4]
We prove that a k-tree can be viewed as a subgraph of a special type of (k + 1)tree that corresponds to a stacked polytope and that these “stacked” (k + 1)-trees admit representations by orthogonal spheres in Rk+1
Among its many interesting properties are that μ is minor monotone and μ(G) 3 if and only if G is planar. These properties are collected in a recent survey by Laszlo Lovasz [17] based on an earlier paper with Hein van der Holst and Alexander Schrijver [28]

Summary

Introduction

Yves Colin de Verdiere’s graph invariant μ is defined as the maximum nullity over a special class of real symmetric matrices [4]. They conjectured that μ(G) + μ(G) |G| − 2 for all graphs G. We obtain one bound for general k-trees and another slightly better one for k-trees that correspond to stacked polytopes The study of these “stacked” k-trees is suggested by recent calculations of μ for chordal graphs by Shaun Fallat and the first author [8], which in turn were based on similar patterns appearing in the study of μ for split graphs by Felix Goldberg and Abraham Berman [9].

Stacked k-Trees

Vector and Sphere Representations

Main Results