Asymptotic Enumeration of Sparse Multigraphs with Given Degrees

Abstract

Let $J$ and $J^*$ be subsets of $\mathbb{N}$ such that $0,1\in J$ and $0\in J^*$. For infinitely many $n$, let ${\boldsymbol{k}}=(k_1,\ldots, k_n)$ 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 $\{1,\ldots, n\}$ with degree sequence given by ${\boldsymbol{k}}$ such that every loop has multiplicity in $J^*$ and every nonloop 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 $k_{\mathrm{max}}$ satisfies $k_{\mathrm{max}} = o(M^{1/3})$. We prove this result using the switching method, building on an asymptotic enumeration of simple graphs with given degrees [B. D. McKay and N. C. Wormald, Combinatorica, 11 (1991), pp. 369--382]. Our application of the switching method introduces a novel way of combining several different switching operations into a single computation.