The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences. We prove that the switch chain is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree @math which satisfies @math , where @math is the sum of the degrees. The mixing time bound obtained is only an order of @math larger than that established in the regular case, where @math is the number of vertices.
@cite_19: We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|1 3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|1 4). In particular, ifk=o(n1 2), the number of regular graphs of degreek and ordern is asymptotically @math Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.
@cite_15: We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d1,…,dn). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0-1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013
@cite_16: 
It is not known whether the corresponding counting problem (exact evaluation of @math ) is @math -complete. There are several results giving asymptotic enumeration formulae for @math under various conditions on @math : see for example @cite_15 @cite_16 @cite_19 and references therein.