Counting loopy graphs with given degrees

Let d = (d1, d2, . . . , dn) be a vector of nonnegative integers. We study the number of symmetric 0-1 matrices whose row sum vector equals d. While previous work has focussed on the case of zero diagonal, we allow diagonal entries to equal 1. Specifically, for D ∈ {1, 2} we define the set GD(d) of all n × n symmetric 0-1 matrices with row sums given by d, where each diagonal entry is multiplied by D when forming the row sum. We obtain asymptotically precise formulae for |GD(d)| in the sparse range (where, roughly, the maximum row sum is o(n1/2)), and in the dense range (where, roughly, the average row sum is proportional to n and the row sums do not vary greatly). The case D = 1 corresponds to enumeration by the usual row sum of matrices. The case D = 2 corresponds to enumeration by degree sequence of undirected graphs with loops but no repeated edges, due to the convention that a loop contributes 2 to the degree of its incident vertex. We also analyse the distribution of the trace of a random element of GD(d), and prove that it is well approximated by a binomial distribution in the dense range, and by a Poisson binomial distribution in the sparse range.