On the Wiener index of generalized Fibonacci cubes and Lucas cubes

• Published in: Discrete Applied Mathematics Vol. 187; pp. 155 - 160
• Main Authors: Klavžar, Sandi, Rho, Yoomi
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 31.05.2015
• Subjects: Generalized Fibonacci cube; Generalized Lucas cube; Hypercube; Isometric embedding; Wiener index; Generalized Lucas cube; Generalized Fibonacci cube; Wiener index; Isometric embedding; Hypercube
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2015.02.002

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(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(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) 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(f↽)) is not isometric in Qd, then for any positive integer k, for almost all dimensions d′ the distance in Qd′(f) (resp. Qd′(f↽)) can exceed the Hamming distance by k.