On Isomorphism Classes of Generalized Fibonacci Cubes

Abstract

The generalized Fibonacci cube $Q_d(f)$ is the subgraph of the $d$-cube $Q_d$ induced on the set of all strings of length $d$ that do not contain $f$ as a substring. It is proved that if $Q_d(f) \cong Q_d(f')$ then $|f|=|f'|$. The key tool to prove this result is a result of Guibas and Odlyzko about the autocorrelation polynomial associated to a binary string. It is also proved that there exist pairs of strings $f, f'$ such that $Q_d(f) \cong Q_d(f')$, where $|f| \ge \frac{2}{3}(d+1)$ and $f'$ cannot be obtained from $f$ by its reversal or binary complementation. Strings $f$ and $f'$ with $|f|=|f'|=d-1$ for which $Q_d(f) \cong Q_d(f')$ are characterized.