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)h(G) of a graph GG is introduced as the Hamming subgraphs counting polynomial. KkKk-derivates ∂kG(k≥2) of a partial Hamming graph are also introduced. It is proved that for a partial Hamming graph GG, ∂h(G)∂xk=h(∂kG). A couple of combinatorial identities involving the coefficients of the Hamming polynomials of Hamming graphs are also proven.