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.
Charles, Z. B., Farber, M., Johnson, C. R., & Kennedy-Shaffer, L. (2013). Nonpositive eigenvalues of the adjacency matrix and lower bounds for Laplacian eigenvalues. Discrete Mathematics, 313(13), 1441-1451.