I have a question in combinatorics. I apologize in advance if some things I say do not make any sense, I don't know much about graph theory or combinatorics. This is a question which came up in a probability problem I'm working on. Let $k=2m$, $1\leq l\leq m$, and $m_1,m_2,\dots,m_l$ be even numbers such that $m_1+m_2+\dots+m_l=k$. What I would like to know is how many nonisomorphic connected graphs there are with $l+1$ distinct vertices, $l$ distinct edges having multiplicities $m_1,m_2,\dots,m_l$.
I would be especially grateful for any papers or books which are related to this problem.