The cube polynomial and its derivatives: The case of median graphs

Abstract

For i ≥, 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, χ) of G is introduced as Σi≥0αi(G)χi. It is shown that any function f with two related, natural properties, is up to the factor f(Q0, χ) 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, χ) = f(∂G, χ) 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, χ + 1) = Σi≥s c(i)(G,χ) /(i-s)! , where certain derivatives of the cube polynomial coincide with well-known invariants of median graphs.