The eccentricity sequences of Fibonacci and Lucas cubes

The Fibonacci cube Γ n is the subgraph of the hypercube 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. The eccentricity of a vertex u , denoted e G ( u ) is the greatest distance between u and any other vertex v in the graph G . For a given vertex u of Γ n we characterize the vertices v such that d Γ n ( u , v ) = e Γ n ( u ) . We then obtain the generating functions of the eccentricity sequences of Γ n and Λ n . As a corollary, we deduce the number of vertices of a given eccentricity.