TY - JOUR

T1 - The isoperimetric constant of the random graph process

AU - Benjamini, Itai

AU - Haber, Simi

AU - Krivelevich, Michael

AU - Lubetzky, Eyal

PY - 2008/1

Y1 - 2008/1

N2 - The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of (Equation Presented), taken over all nonempty subsets S ⊂ V(G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, G̃(t), is a sequence of (Equation Presented) graphs, where G̃(0) is the edgeless graph on n vertices, and G̃(t) is the result of adding an edge to G(t- 1), uniformly distributed over all the missing edges. The authors show that in almost every graph process i(G̃(t)) equals the minimal degree of G̃(t) as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to 1/2, its final value.

AB - The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of (Equation Presented), taken over all nonempty subsets S ⊂ V(G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, G̃(t), is a sequence of (Equation Presented) graphs, where G̃(0) is the edgeless graph on n vertices, and G̃(t) is the result of adding an edge to G(t- 1), uniformly distributed over all the missing edges. The authors show that in almost every graph process i(G̃(t)) equals the minimal degree of G̃(t) as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to 1/2, its final value.

KW - Conductance

KW - Isoperimetric constant

KW - Minimal degree

KW - Random graph process

UR - http://www.scopus.com/inward/record.url?scp=38049108422&partnerID=8YFLogxK

U2 - 10.1002/rsa.20171

DO - 10.1002/rsa.20171

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AN - SCOPUS:38049108422

SN - 1042-9832

VL - 32

SP - 101

EP - 114

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 1

ER -