In random graphs with feasible degree sequences containing a giant component, the giant is unique whp with explicit diameter and mixing-time bounds that are often tight, along with size and diameter controls on smaller components.
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Diameters and mixing times for giant components of random graphs with given degrees
In random graphs with feasible degree sequences containing a giant component, the giant is unique whp with explicit diameter and mixing-time bounds that are often tight, along with size and diameter controls on smaller components.