On the sizes of extended Fibonacci cubes

Hypercube is a popular interconnection network whose size must be a power of 2. Several interconnection networks have been proposed that do not suffer this limitation. Among them the extended Fibonacci cubes are based on the same sequence of the Fibonacci cubes and share many appealing structural properties. In this paper, we show how Extended Fibonacci Cubes can be seen as (Cartesian) product graphs whose components are hypercubes and Fibonacci Cubes. By exposing this property, we prove a conjecture that there are no distinct Extended Fibonacci Cubes (except the trivial ones) with the same number of nodes. Our result further validates the motivations behind the proposal of this interconnection network as a flexible alternative to hypercubes.