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Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of anyundirected graph with $n$ vertices or more has at least $k$ nonpositiveeigenvalues. 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, weprove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangularnumber. This implies new lower bounds for eigenvalues of Laplacian matrices:the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largestdegree, which generalizes some prior results.

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