Hamming polynomials and their partial derivatives (Q875053)

A Hamming graph is a Cartesian product of complete graphs; its isometric subgraphs are partial Hamming graphs. The Hamming polynomial of a graph is its Hamming subgraphs counting polynomial. It is proven that any partial derivative of the Hamming polynomial of a partial Hamming graph is itself a Hamming polynomial of a graph, the \(K_k\)-derivate, for which an explicit construction is given. Further two combinatorial identities are derived for the diagonal Hamming polynomial coefficients of any power of a complete graph.