Let G=(V,E) be an undirected graph. Given ℓ∈\Z, let m:V→\Z+ be a non-negative integer valued function satisfying
In this section, we discuss related graph two-sample problems, and comment on practical two-sample tests.
This will be a simple consequence of basic properties of the mixed volume. For this, let us recall a well-known alternative formula (see e.g. ).
We first treat a self-intersection as a singularity and ‘pull’ information from it in a manner analogous to the proof of Simon’s monotonicity formula . See also Theorem 6 in  and the appendix of .
For the remainder, we investigate an universal upper bound on the fiber dimension by generalizing our embedding in Proposition 2.3 into the simplex. A priori, a move in a Markov basis give rise to distinguished edges in a fiber graph. The next defin…
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