Cubes polynomial and its derivatives

Let i (G) be the number of induced i-cubes of a graph G. Then the cubes polynomial c(G; x) of G is introduced as P i0 i (G)x i. It is shown that any function f with two related, natural properties, is up to the factor f(K 1 ; 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 f 0 (G; x) = f(@ G; x) provided that f(G; 0) = jV (G)j. Several relations that generalize many previous results for median graphs are also given. For instance, for any s 0 we have c (s)