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Asymptotic enumeration of sparse multigraphs with given degrees

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.