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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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: