On the spread of supercritical random graphs

Details On the spread of supercritical random graphs On the spread of supercritical random graphs

2009-02-06

arxiv.org/abs/0902.1156v1

Mathematics

Probability

17 pages

Scientific paper

The spread of a connected graph G was introduced by Alon Boppana and Spencer (1998) and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on $V(G)$. We investigate the spread of a variety of random graphs, in particular the random regular graphs G(n,d), d >= 3, and Erdos-Renyi random graphs G_{n,p} in the supercritical range p>1/n. We show that if p=c/n with c>1 fixed then with high probability the spread is bounded, and prove similar statements for G(n,d), d >= 3. We also prove lower bounds on the spread in the barely supercritical case p-1/n = o(1). Finally, we show that for d large the spread of G(n,d) becomes arbitrarily close to that of the complete graph K_n.

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