The degree sequence of Fibonacci and Lucas cubes

Sandi Klavzar (1) , Michel Mollard (2) , Marko Petkovsek (1)

1 FMF - Faculty of Mathematics and Physics [Ljubljana]
2 IF - Institut Fourier

The Fibonacci cube $\Gamma_n$ is the subgraph of the $n$-cube induced by the binary strings that contain no two consecutive 1's. The Lucas cube $\Lambda_n$ is obtained from $\Gamma_n$ by removing vertices that start and end with 1. It is proved that the number of vertices of degree $k$ in $\Gamma_n$ and $\Lambda_n$ is $\sum_{i = 0}^k \binom{n-2i}{k-i} \binom{i+1}{n-k-i+1}$ and $\sum_{i=0}^{k} \left[2\binom{i}{2i+k-n} \binom{n-2i-1}{k-i} + \binom{i-1}{2i+k-n} \binom{n-2i}{k-i}\right]$, respectively. Both results are obtained in two ways, since each of the approaches yields additional results on the degree sequences of these cubes. In particular, the number of vertices of high resp.\ low degree in $\Gamma_n$ is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of $\Gamma_n$ and $\Lambda_n$ are easily computed

Mots clés: Fibonacci cube; Lucas cube; Degree sequence; Generating function

Domaines: Combinatoire [math.CO]

Sandi Klavzar, Michel Mollard, Marko Petkovsek. The degree sequence of Fibonacci and Lucas cubes. Discrete Mathematics, 2011, 311, pp.1310-1322. ⟨hal-00536061⟩