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

Zachary B. Charles, Miriam Farber, Charles R. Johnson, Lee Kennedy-Shaffer
2013 Discrete Mathematics  
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
more » ... Laplacian matrices: the kth largest eigenvalue is bounded from below the NPO(k)th largest degree, which generalizes some prior results.
doi:10.1016/j.disc.2013.03.010 fatcat:fdbwhfdvr5ds5h2ycmez7xwypy