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

Abstract

For $i\geq 0$, the $i$-cube $Q_i$ is the graph on $2^i$ vertices representing $0/1$ tuples of length $i$, where two vertices are adjacent whenever the tuples differ in exactly one position. (In particular, $Q_0 = K_1$.) Let $\alpha_i(G)$ be the number of induced $i$-cubes of a graph $G$. Then the cube polynomial $c(G,x)$ of $G$ is introduced as $\sum_{i\geq 0} \alpha_i(G) x^i$. It is shown that any function $f$ with two related, natural properties, is up to the factor $f(Q_0,x)$ the cube polynomial. The derivation $\partial\, G$ of a median graph $G$ is introduced and it is proved that the cube polynomial is the only function $f$ with the property $f'(G,x)= f(\partial\, G, x)$ 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 $s\geq 0$ we have $c^{(s)}(G,x+1) = \sum_{i\geq s}\, {{c^{(i)}(G,x)}\over {(i-s)!}}\,,$ where certain derivatives of the cube polynomial coincide with well-known invariants of median graphs.