On the Wiener index of generalized Fibonacci cubes and Lucas cubes

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