Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues

Let NPO(k) be the smallest number n such that the adjacency matrix of any undirected graph with n vertices or more has at least k nonpositive eigenvalues. We show that NPO(k) is well-defined and prove that the values of NPO(k) for k = 1, 2, 3, 4, 5 are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all k >= 5, R(k, k + 1) >= NPO(k) > T-k, in which R(k, k + 1) is the Ramsey number for k and k + 1, and T-k is the kth triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the kth largest eigenvalue is bounded from below the NPO(k)th largest degree, which generalizes some prior results. (C) 2013 Elsevier B.V. All rights reserved.