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I have tried to count the number of simple labelled bipartite graphs $G_{n,m}$ with $k$ edges such that $d_1$ vertices have degree 1.

Has this problem been studied?

So far the only related paper I found shows a generating function to count all simple labelled graphs where monovalent vertices are represented by u:

$M(u,x) = \sum_{k \geq 0} \frac{x^k}{k!}2^{{k\choose 2}}(e^{ux})^{k}e^{u^2x^2/2}$

This paper also has some results on bipartite graphs, but as I'm not familiar with generating functions, I don't know how to adapt these results to my original problem. Any help would be appreciated.

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    $\begingroup$ do you want any bipartite graphs or connected bipartite graphs? $\endgroup$
    – vidyarthi
    Commented Aug 12, 2020 at 9:24
  • $\begingroup$ any bipartite graphs $\endgroup$
    – Helene
    Commented Aug 12, 2020 at 23:33

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