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?

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
