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

Let N P O ( 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 N P O ( k ) is well-defined and prove that the values of N P O ( 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 ) ≥ N P O ( k ) > T k , in which R ( k , k + 1 ) is the Ramsey number for k and k + 1 , and T k is the k th triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the k th largest eigenvalue is bounded from below the N P O ( k ) th largest degree, which generalizes some prior results.