Lack of Spectral Gap and Hyperbolicity in Asymptotic Erdös-Renyi Random Graphs

Details Lack of Spectral Gap and Hyperbolicity in Asymptotic Erdös-Renyi Random Graphs Lack of Spectral Gap and Hyperbolicity in Asymptotic Erdös-Renyi Random Graphs

2010-09-28

arxiv.org/abs/1009.5700v1

Mathematics

Probability

13 pages

Scientific paper

In this work, we prove the absence of a spectral gap for the normalized
Laplacian of the Erd\"os-Renyi random graph $G(n,p)$ when $p=\frac{d}{n}$ for
$d>1$ as $n\to\infty$. We also prove that for any positive $\delta$ the
Erd\"os-Renyi random graph has a positive probability of containing
$\delta$-fat triangles as $n\to\infty$.

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