Prove complement of this 12-vertex graph is a tree Let G12 be a simple graph of 12 vertices, and H12 its complement. It is known that G12 has 7 vertices of degree 10, 2 vertices of degree 9, 1 vertex of degree 8 and 2 vertices of degree 7. Prove or disprove H12 is a tree.

This is a matter of whether you can create a complement graph that is disconnected or not: when the number of edges is 1 less than the number of nodes as in your $Q$, the complement graph is a tree iff it is connected. 
Whenever you can come up with a complement graph that is a tree 
with these numbers, you can modify it to be a disconnected graph with the same 
number of nodes and edge distribution among nodes:
take two leaves in the tree where they have different parents. remove these leaves 
and connect them with an edge so that the two are connected with one another 
and disconnected from the rest. Put an edge connecting the parents of these 
two leaves. The disturbed nodes preserve their degree in the resulting graph and 
the resulting graph is disconnected with the same number of edges. A disconnected graph with 11 edges can be turned into a tree in a similar way.
Your $Q$ has no solution for disproving or proving it on one side. the resulting 
graph can be either. 
