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)>Tk, in which R(k,k+1) is the Ramsey number for k and k+1, and Tk 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.