Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the spanning trees of the hypercube and other products of graphs   We give two combinatorial proofs of an elegant product formula for the number
of spanning trees of the $n$-dimensional hypercube. The first proof is based on
the assertion that if one chooses a uniformly random rooted spanning tree of
the hypercube and orient each edge from parent to child, then the parallel
edges of the hypercube get orientations which are independent of one another.
This independence property actually holds in a more general context and has
intriguing consequences. The second proof uses some "killing involutions" in
order to identify the factors in the product formula. It leads to an
enumerative formula for the spanning trees of the $n$-dimensional hypercube
augmented with diagonals edges, counted according to the number of edges of
each type. We also discuss more general formulas, obtained using a matrix-tree
approach, for the number of spanning trees of the Cartesian product of complete
graphs.

14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph