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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 at 23:23

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There are none. By the hand shaking lemma we know that the number of degrees of odd degree must be even.

There are 5 vertices with odd degrees in your graph, these are the ones with degrees:

1,1,1,1,5

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