  We investigate the number of symmetric matrices of non-negative integers with
zero diagonal such that each row sum is the same. Equivalently, these are zero
diagonal symmetric contingency tables with uniform margins, or loop-free
regular multigraphs. We determine the asymptotic value of this number as the
size of the matrix tends to infinity, provided the row sum is large enough. We
conjecture that our answer is valid for all row sums.
