Fibonacci cube
The Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in number theory. Mathematically they are similar to the hypercube graphs, but with a Fibonacci number of vertices. Fibonacci cubes were first explicitly defined in Hsu (1993) in the context of interconnection topologies for connecting parallel or distributed systems. They have also been applied in chemical graph theory.
The Fibonacci cube may be defined in terms of Fibonacci codes and Hamming distance, independent sets of vertices in path graphs, or via distributive lattices.
Like the hypercube graph, the vertices of the Fibonacci cube of order n may be labeled with bitstrings of length n, in such a way that two vertices are adjacent whenever their labels differ in a single bit. However, in a Fibonacci cube, the only labels that are allowed are bitstrings with no two consecutive 1 bits. There are Fn + 2 labels possible, where Fn denotes the nth Fibonacci number, and therefore there are Fn + 2 vertices in the Fibonacci cube of order n.
The nodes of such a network may be assigned consecutive integers from 0 to Fn + 2 − 1; the bitstrings corresponding to these numbers are given by their Zeckendorf representations.
The Fibonacci cube of order n is the simplex graph of the complement graph of an n-vertex path graph. That is, each vertex in the Fibonacci cube represents a clique in the path complement graph, or equivalently an independent set in the path itself; two Fibonacci cube vertices are adjacent if the cliques or independent sets that they represent differ by the addition or removal of a single element. Therefore, like other simplex graphs, Fibonacci cubes are median graphs and more generally partial cubes. The median of any three vertices in a Fibonacci cube may be found by computing the bitwise majority function of the three labels; if each of the three labels has no two consecutive 1 bits, the same is true of their majority.
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