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
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
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 -