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Isolated vertices in the Erdős-Rényi model

Let

N

be the number of isolated vertices in

G (n, p)

. Indicator variables give

E N = n (1 - p)^{n - 1}

and, because a specified pair is simultaneously isolated precisely when the

2 n - 3

incident edges are absent,

E (N^{2}) = n (1 - p)^{n - 1} + n (n - 1) (1 - p)^{2 n - 3} .

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