  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 $d_{\max}$ which satisfies
$3\leq d_{\max}\leq \frac{1}{4}\, \sqrt{M}$, where $M$ is the sum of the
degrees. The mixing time bound obtained is only an order of $n$ larger than
that established in the regular case, where $n$ is the number of vertices.
