On the largest component of a random graph with a subpower-law degree sequence in a subcritical phase

A uniformly random graph on n vertices with a fixed degree sequence, obeying a . subpower law, is studied. It is shown that, for .>3, in a subcritical phase with high probability the largest component size does not exceed n1/.+.n, .n=O(ln.ln.n/ln.n), 1/. being the best power for this random graph. This is similar to the best possible n1/(..1) bound for a different model of the random graph, one with independent vertex degrees, conjectured by Durrett, and proved recently by Janson.