Given a set D of nonnegative integers, we derive the asymptotic number of graphs with a givenvnumber of vertices, edges, and such that the degree of every vertex is in D. This generalizes existing results, such as the enumeration of graphs with a given minimum degree, and establishes new ones, such as the enumeration of Euler graphs, i.e. where all vertices have an even degree. Those results are derived using analytic combinatorics.
@cite_5: Counting formulae are obtained for the numbers of labelled (p,q)-multigraphs with any even number n of points of odd degree and p-n points of even degree. Similar results are obtained for multigraphs of strength s, and in this case for the sum over the number q of lines. These sums are much less simple if s is even than if s is odd. However, it is shown that for s fixed, and p -> , the asymptotic value does not depend on the parity of s.
@cite_16: 
@cite_11: Abstract Asymptotics are obtained for the number of n × n symmetric non-negative integer matrices subject to the following constraints: (i) each row sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled graphs.
Other related problems include the enumeration of graphs with a given degree sequence ( @cite_11 ), the enumeration of symmetric matrices with nonnegative coefficients and constant row sum ( @cite_16 ), and the enumeration of graphs with degree parities, investigated by @cite_5 .