Cubes polynomial and its derivatives

Let αi(G) be the number of induced i-cubes of a graph G. Let c(G,x) be the generating function of the sequence (αi(G))∞i=0) that is, c(G,x) = ∑i≥0αi(G)xi For finite graphs G, c(G, x) is a polynomial and we call it the cubes polynomial of G. shown that any function f with two related, natural properties, is up to the factor f(K1, x) the cubes polynomial. The derivation ∂G of a median graph G is also introduced and it is proved that the cubes polynomial is the only function f with the property fʹ(G,x) = f(∂G,x) provided that f(G, 0) = ∣V(G)∣. Several relations that generalize many previous results for median graphs are also given.