1009.5700 (Onuttom Narayan et al.)

Lack of Hyperbolicity in Asymptotic Erdös--Renyi Sparse Random Graphs    [PDF]

Onuttom Narayan, Iraj Saniee, Gabriel H. Tucci

In this work we prove that the giant component of the Erd\"os--Renyi random graph $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $\delta$-hyperbolic for any positive $\delta$ with probability tending to one as $n\to\infty$. As a corollary we provide an alternative proof that the giant component of $G(n,c/n)$ when c>1 has zero spectral gap almost surely as $n\to\infty$.

View original: http://arxiv.org/abs/1009.5700