Originally Posted by
Prove It From the handshaking lemma
$\displaystyle \sum{\textrm{deg}\,(v)} = 2|E|$
Since there are $\displaystyle 33$ edges, that means $\displaystyle \sum{\textrm{deg}\,(v)} = 66$.
That means that $\displaystyle 7m + 8n = 66$, where $\displaystyle m$ is the number of vertices with degree $\displaystyle 7$ and $\displaystyle n$ is the number of vertices with degree $\displaystyle 8$.
By inspection, $\displaystyle m = 6$ and $\displaystyle n = 3$.
So there are $\displaystyle 9$ vertices.