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arXiv · 1010.4269

Minimum vertex covers and the spectrum of the normalized Laplacian on trees

Hao Chen·Jürgen Jost
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Abstract

We show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.

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Hao Chen, Jürgen Jost. 2010-10-20. Minimum vertex covers and the spectrum of the normalized Laplacian on trees. https://doi.org/10.1016/j.laa.2012.04.005

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