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

For i ≥ 0, the i-cube Qi is the graph on 2i vertices representing 0/1 tuples of length i, where two vertices are adjacent whenever the tuples differ in exactly one position. (In particular, Q0 = K1.) Let αi(G) be the number of induced i-cubes of a graph G. Then the cube polynomial c(G,x) of G is introduced as ∑ i≥0 αi(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 ′(G,x) = f(∂ G, x) provided that f(G, 0) = |V (G)|. As 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 ≥ 0 we have c(s)(G,x + 1) = ∑i≥s c(i)(G,x) (i−s)! , where certain derivatives of the cube polynomial coincide with well-known invariants of median graphs. ∗Supported by the Ministry of Education, Science and Sport of Slovenia under the grant Z1-30730101-01. †Supported by the same Ministry under the grant 101–504. ‡Supported by the same Ministry under the grant Z1-3219. the electronic journal of combinatorics 10 (2003), #R3 1