The degree sequence of Fibonacci and Lucas cubes

Abstract

The Fibonacci cube Γ n is the subgraph of the n -cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube Λ n is obtained from Γ n by removing vertices that start and end with 1. It is proved that the number of vertices of degree k in Γ n and Λ n is ∑ i = 0 k ( n − 2 i k − i ) ( i + 1 n − k − i + 1 ) and ∑ i = 0 k [ 2 ( i 2 i + k − n ) ( n − 2 i − 1 k − i ) + ( i − 1 2 i + k − n ) ( n − 2 i k − i ) ] , 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 Γ n is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of Γ n and Λ n are easily computed.