# How many graphs with vertex degrees (1, 1, 1, 1, 2, 4, 5, 6, 6) are there?

How many graphs with vertex degrees (1, 1, 1, 1, 2, 4, 5, 6, 6) are there? Assuming that all vertices and edges are labelled. I know there's a long way to do it by drawing all of them and count. Is there a quicker, combinatoric way?

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The sum of the degrees must be even for there to be any, because each edge adds $2$ to the sum of the degrees. –  mjqxxxx Dec 1 '12 at 22:05
There is a software package nauty, by Brendan McKay that can automate the solution of feasible versions of such problems. See in particular the tool geng of that package. –  hardmath Dec 1 '12 at 22:09
nauty is used for graph canonical labeling. It comes with a package "geng" which can generate graphs upto isomorphism under certain conditions, but degree sequence is not one of them. –  Douglas S. Stones Apr 16 '13 at 23:23