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{\bf Bo\v{s}tjan Bre\v{s}ar, Sandi Klav\v{z}ar and Riste \v Skrekovski}
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{\bf The Cube Polynomial and its Derivatives: the Case of Median Graphs}
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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.


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