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The published version of this part of Li's thesis is I. M. Gessel and J. Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 691-612. But the formula we give in this paper for the cycle index series for bicolored graphs isn't really new, I don't think; certainly the formula for unlabeled bicolored graphs that you get from it isn't new (and this isn't the point of our paper). But the paper does have references to earlier work on this topic.
Not really relevant to this question, but if $a$ and $b$ are integers and $\binom{a+bi}{k}= c+di$, where $c$ and $d$ are rational, then every prime dividing the denominator of $c$ or $d$ is congruent to 2 or 3 modulo 4.
It's unlikely that you can find any formulas for these numbers that can't be derived easily from the generating function. One thing that you can do is find a fairly simple recurrence by differentiating the generating function.
The variation in which we count unlabeled bicolored graphs, i.e, isomorphism classes of graphs with m red and n blue vertices, where every edge connects a blue vertex to a red vertex, and where isomorphisms must preserve the colors of the vertices, is much easier, and can be solved by a straightforward application of Polya's theorem (or Burnside's lemma).
This paper is about spanning trees, not spanning graphs. The problem is a variation of the problem of counting unlabeled bipartite graphs and it seems likely that it could be solved using the methods that can be used to count bipartite graphs. See, for example, Frank Harary and Geert Prins, Enumeration of bicolourable graphs, Canad. J. Math. 15 (1963), 237–248 and Phil Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), 49–57.
