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Let  G  be a  d ‐regular graph  G  on  n  vertices. Suppose that the adjacency matrix of  G  is such that the eigenvalue λ which is second largest in absolute value satisfies λ =  o ( d ). Let  G   p   with  p  = α/ d  be obtained from  G  by including each edge of  G  independently with probability  p . We show that if α < 1, then whp the maximum component size of  G   p   is  O (log  n ) and if α > 1, then  G   p   contains a unique giant component of size Ω( n ), with all other components of size  O (log  n ). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004

The emergence of a giant component in random subgraphs of pseudo‐random graphs