  Let J and J* be subsets of Z+ such that 0,1\in J and 0\in J*. For infinitely
many n, let k=(k_1,..., k_n) be a vector of nonnegative integers whose sum M is
even. We find an asymptotic expression for the number of multigraphs on the
vertex set {1,..., n} with degree sequence given by k, such that every loop has
multiplicity in J* and every non-loop edge has multiplicity in J. Equivalently,
these are symmetric integer matrices with values J* allowed on the diagonal and
J off the diagonal. Our expression holds when the maximum degree K satisfies K
= o(M^(1/3)). We prove this result using the switching method, building on an
asymptotic enumeration of simple graphs with given degrees (McKay and Wormald,
1991). Our application of the switching method introduces a novel way of
combining several different switching operations into a single computation.
