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arXiv · 1205.2321

On the spectral density function of the Laplacian of a graph

Herbert Koch·Wolfgang Lueck
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Abstract

Let X be a finite graph. Let E be the number of its edges and d be its degree. Denote by F_1(X) its first spectral density function which counts the number of eigenvalues less or equal to lambda^2 of the associated Laplace operator. We prove the estimate F_1(X)(lambda) - F_1(X)(0) le 2 cdot E cdot d cdot lambda for 0 le lambda < 1. We explain how this gives evidence for conjectures about approximating Fuglede-Kadison determinants and L^2-torsion.

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Herbert Koch, Wolfgang Lueck. 2013-06-03. On the spectral density function of the Laplacian of a graph. https://arxiv.org/abs/1205.2321

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