Let $G$ be a ribbon graph (sometimes called fat graph) with $v$ vertices and $e$ edges. Furthermore each vertex is of degree $d$.
Q) What is the number of $G$ with the above properties? I mean does there exist a closed formula for the number of such $G$ in terms of $v$, $e$, and $d$.
I don't know any reference or result in this area. I searched but I could not find anything of this type. Any reference, link, suggestion will be helpful.
Thanks in advance.