The isoperimetric constant of the random graph process

• Authors: Itai Benjamini;Simi Haber;Michael Krivelevich;Eyal Lubetzky
• Venue: Random Structures & Algorithms
• Year: 2008

The isoperimetric constant of a graph G on nvertices, i(G), is the minimum of |∂S| / |S|,taken over all nonempty subsets S ⊂ V (G)of size at most n/2, where ∂S denotes theset of edges with precisely one end in S. A random graphprocess on n vertices, G(t), is a sequence of $n\choose 2$ graphs, where $\tilde{G}(0)$ is the edgeless graph onn vertices, and $\tilde{G}(t)$ is the result of adding anedge to $\tilde{G}(t-1)$, uniformly distributed over all themissing edges. The authors show that in almost every graph process$i(\tilde{G}(t))$ equals the minimal degree of $\tilde{G}(t)$ aslong as the minimal degree is o(log n). Furthermore,it is shown that this result is essentially best possible, bydemonstrating that along the period in which the minimum degree istypically ˜(log n), the ratio between theisoperimetric constant and the minimum degree falls from 1 to$1\over 2$, its final value. © 2007 Wiley Periodicals, Inc.Random Struct. Alg., 2008