  We study and derive identities for the multi-variate independence polynomials
from the perspective of heaps theory. Using the inversion formula and the
combinatorics of partially commutative algebras we show how the multi-variate
version of Godsil type identity as well as the fundamental identity can be
obtained from weight preserving bijections. Finally, we obtain a new
multi-variate identity involving connected bipartite subgraphs similar to the
Christoffel-Darboux type identities obtained by Bencs.
