The eccentricity sequences of Fibonacci and Lucas cubes

The Fibonacci cube ΓnΓn is the subgraph of the hypercube 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. The eccentricity of a vertex uu, denoted eG(u)eG(u) is the greatest distance between uu and any other vertex vv in the graph GG. For a given vertex uu of ΓnΓn we characterize the vertices vv such that dΓn(u,v)=eΓn(u)dΓn(u,v)=eΓn(u). We then obtain the generating functions of the eccentricity sequences of ΓnΓn and ΛnΛn. As a corollary, we deduce the number of vertices of a given eccentricity.