The isoperimetric constant of a graph G on n vertices, i(G), is the
minimum of β£Sβ£β£βSβ£β, taken over all nonempty subsets
SβV(G) of size at most n/2, where βS denotes the set of
edges with precisely one end in S. A random graph process on n vertices,
G(t), is a sequence of (2nβ) graphs, where
G(0) is the edgeless graph on n vertices, and
G(t) is the result of adding an edge to G(tβ1),
uniformly distributed over all the missing edges. We show that in almost every
graph process i(G(t)) equals the minimal degree of
G(t) as long as the minimal degree is o(logn). Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically Ξ(logn), the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value