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

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

2011-08-24

arxiv.org/abs/1108.4810v2

Mathematics

Combinatorics

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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