  A finite non-increasing sequence of positive integers $d = (d_1\geq
\cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$
with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that
case we say that the graph $G$ realizes the degree sequence $d$. We show that
the exact number of labeled graphs that realize a fixed degree sequence
satisfies a simple recurrence relation. Using this relation, we then obtain a
recursive algorithm for the exact count. We also show that in the case of
regular graphs the complexity of our algorithm is better than the complexity of
the same enumeration that uses generating functions.
