On the Wiener index of generalized Fibonacci cubes and Lucas cubes

The generalized Fibonacci cube Qd(f)Qd(f) is the graph obtained from the dd-cube QdQd by removing all vertices that contain a given binary word ff as a factor; the generalized Lucas cube Qd(f↽) is obtained from QdQd by removing all the vertices that have a circulation containing ff as a factor. In this paper the Wiener index of Qd(1s)Qd(1s) and the Wiener index of Qd(1s↽) are expressed as functions of the order of the generalized Fibonacci cubes. For the case Qd(111)Qd(111) a closed expression is given in terms of Tribonacci numbers. On the negative side, it is proved that if for some dd, the graph Qd(f)Qd(f) (or Qd(f↽)) is not isometric in QdQd, then for any positive integer kk, for almost all dimensions d′d′ the distance in Qd′(f)Qd′(f) (resp. Qd′(f↽)) can exceed the Hamming distance by kk.