Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting spanning trees in a complete bipartite graph which contain a
  given spanning forest   In this article, we extend Moon's classic formula for counting spanning trees
in complete graphs containing a fixed spanning forest to complete bipartite
graphs. Let $(X,Y)$ be the bipartition of the complete bipartite graph
$K_{m,n}$ with $|X|=m$ and $|Y|=n$. We prove that for any given spanning forest
$F$ of $K_{m,n}$ with components $T_1,T_2,\ldots,T_k$, the number of spanning
trees in $K_{m,n}$ which contain all edges in $F$ is equal to $$ \frac
1{mn}\left(\prod_{i=1}^k (m_in+n_im)\right) \left
(1-\sum_{i=1}^{k}\frac{m_in_i}{m_in+n_im}\right ), $$ where $m_i=|V(T_i)\cap
X|$ and $n_i=|V(T_i)\cap Y|$ for $i=1,2,\ldots,k$.

15
The categories of this paper are: nucl-ex
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph