2 research outputs found

    Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues

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    Let NPO(k)NPO(k) be the smallest number nn such that the adjacency matrix of any undirected graph with nn vertices or more has at least kk nonpositive eigenvalues. We show that NPO(k)NPO(k) is well-defined and prove that the values of NPO(k)NPO(k) for k=1,2,3,4,5k=1,2,3,4,5 are 1,3,6,10,161,3,6,10,16 respectively. In addition, we prove that for all kβ‰₯5k \geq 5, R(k,k+1)β‰₯NPO(k)>TkR(k,k+1) \ge NPO(k) > T_k, in which R(k,k+1)R(k,k+1) is the Ramsey number for kk and k+1k+1, and TkT_k is the kthk^{th} triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the kk-th largest eigenvalue is bounded from below by the NPO(k)NPO(k)-th largest degree, which generalizes some prior results.Comment: 23 pages, 12 figure
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