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 Verdi`ere, 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 dimL ≤ 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 dimL = 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 6= j and ij 6∈ E (and no restriction on the diagonal), having exactly one negative eigenvalue, satisfies the Strong Arnol’d Hypothesis.