The degree sequence of Fibonacci and Lucas cubes

The Fibonacci cube ΓnΓn is the subgraph of the nn-cube induced by the binary strings that contain no two consecutive 1’s. The Lucas cube ΛnΛn is obtained from ΓnΓn by removing vertices that start and end with 1. It is proved that the number of vertices of degree kk in ΓnΓn and ΛnΛn is ∑i=0k(n−2ik−i)(i+1n−k−i+1) and ∑i=0k[2(i2i+k−n)(n−2i−1k−i)+(i−12i+k−n)(n−2ik−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 ΓnΓn is expressed as a sum of few terms, and the generating functions are given from which the moments of the degree sequences of ΓnΓn and ΛnΛn are easily computed.