Hamming polynomials and their partial derivatives

Hamming graphs are Cartesian products of complete graphs and partial Hamming graphs are their isometric subgraphs. The Hamming polynomial h ( G ) of a graph G is introduced as the Hamming subgraphs counting polynomial. K k -derivates ∂ k G ( k ≥ 2 ) of a partial Hamming graph are also introduced. It is proved that for a partial Hamming graph G , ∂ h ( G ) ∂ x k = h ( ∂ k G ) . A couple of combinatorial identities involving the coefficients of the Hamming polynomials of Hamming graphs are also proven.