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For i 0, the i-cube Qi is the graph on 2 i vertices representing 0/1 tuples of length i, where two vertices are adjacent whenever the tuples dier in exactly one position. (In particular, Q0 = K1.) Let i(G) be the number of induced i-cubes of ag raphG. Then the cube polynomial c(G, x )o fG 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(Q0 ,x ) the cube polynomial. The derivation @G of a median graph G is introduced and it is proved that the cube polynomial is the only function f with the property f 0 (G, x )= f( @G , x) provided that f(G, 0) = |V (G)| .A s the main application of the new concept, several relations that widely generalize previous such results for median graphs are proved. For instance, it is shown that for any s 0w e havec (s) (G, x +1 ) = P is c (i)(G,x)

The Cube Polynomial and its Derivatives: the Case of Median Graphs