  In this paper we introduce the concept of characteristic number that are
proven to be useful in the study of the combinatorics of graph cohomology. We
claim that it is a good combinatorial counterpart for geometric Betti numbers.
We then use this concept and tools built along the way to study Hamiltonian GKM
manifolds whose moment maps are in general position. We prove some connectivity
properties of the their GKM graphs and show an upper bound of their second
Betti numbers, which allows us to conclude that these manifolds, in the case of
dimension 8 and 10, have non-decreasing even Betti numbers up to half
dimension.
