On the expansion of the giant component in percolated (n,d,λ) graphs

Details On the expansion of the giant component in percolated (n,d,λ) graphs On the expansion of the giant component in percolated (n,d,λ) graphs

2005-09-12

arxiv.org/abs/math/0509253v2

Mathematics

Probability

Scientific paper

Let d \geq d_0 be a sufficiently large constant. A (n,d,c \sqrt{d}) graph G is a d-regular graph over n vertices whose second largest (in absolute value) eigenvalue is at most c \sqrt{d}. For any 0 < p < 1, G_p is the graph induced by retaining each edge of G with probability p. It is known that for p > \frac{1}{d} the graph G_p almost surely contains a unique giant component (a connected component with linear number vertices). We show that for p \geq frac{5c}{\sqrt{d}} the giant component of G_p almost surely has an edge expansion of at least \frac{1}{\log_2 n}.

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