On the strong Arnol'd hypothesis and the connectivity of graphs

Abstract

In the definition of the graph parameters μ(G) and ν(G), introduced by Colin de Verdière, and in the definition of the graph parameter ξ(G), introduced by Barioli, Fallat, and Hogben, a transversality condition is used, called the Strong Arnol'd Hypothesis. In this paper, we define the Strong Arnol'd Hypothesis for linear subspaces L ⊆ ℝn with respect to a graph G = (V,E), with V = {1, 2,., n}. We give a necessary and sufficient condition for a linear subspace L ⊆ ℝn with dim L ≤ 2 to satisfy the Strong Arnol'd Hypothesis with respect to a graph G, and we obtain a sufficient condition for a linear subspace L ⊆ ℝn with dim L ≤ 3 to satisfy the Strong Arnol'd Hypothesis with respect to a graph G. We apply these results to show that if G = (V,E) with V = {1, 2,., n} is a path, 2-connected outerplanar, or 3-connected planar, then each real symmetric n × n matrix M = [mi,j] with mi,j < 0 if ij ∈ E and mi,j = 0 if i ≠ j and ij ∈E (and no restriction on the diagonal), having exactly one negative eigenvalue, satisfies the Strong Arnol'd Hypothesis.