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

Mathematics – Combinatorics

Scientific paper

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Scientific paper

Let $NPO(k)$ be the smallest number $n$ such that all the graphs of order $n$ or more have at least $k$ non-positive 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 give an upper and lower bound for $NPO(k)$ for each $k$. This yields a new lower bound for the Laplacian eigenvalues: The $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree.

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