Lack of Hyperbolicity in Asymptotic Erdös--Renyi Sparse Random Graphs

In this work we prove that the giant component of the Erdös--Renyi random graph $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $δ$-hyperbolic for any positive $δ$ 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 $n\to\infty$.

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