  Hodge's formula represents the gravitational MHV amplitude as the determinant
of a minor of a certain matrix. When expanded, this determinant becomes a sum
over weighted trees, which is the form of the MHV formula first obtained by
Bern, Dixon, Perelstein, Rozowsky and rediscovered by Nguyen, Spradlin,
Volovich and Wen. The gravity MHV amplitude satisfies the Britto, Cachazo, Feng
and Witten recursion relation. The main building block of the MHV amplitude,
the so-called half-soft function, satisfies a different, Berends-Giele-type
recursion relation. We show that all these facts are illustrations to a more
general story.
  We consider a weighted Laplacian for a complete graph of n vertices. The
matrix tree theorem states that its diagonal minor determinants are all equal
and given by a sum over spanning trees. We show that, for any choice of a
cocycle on the graph, the minor determinants satisfy a Berends-Giele as well as
Britto-Cachazo-Feng-Witten type recursion relation. Our proofs are purely
combinatorial.
