On the Hyperbolicity of Random Graphs

Dieter Mitsche, Paweł Prałat

Abstract


Let $G=(V,E)$ be a connected graph with the usual (graph) distance metric $d:V \times V \to \mathbb{N} \cup \{0 \}$. Introduced by Gromov, $G$ is $\delta$-hyperbolic if for every four vertices $u,v,x,y \in V$, the two largest values of the three sums $d(u,v)+d(x,y)$, $d(u,x)+d(v,y)$, $d(u,y)+d(v,x)$ differ by at most $2\delta$. In this paper, we determine precisely the value of this hyperbolicity for most binomial random graphs.


Keywords


random graphs, hyperbolicity, diameter

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