On the Wiener index of generalized Fibonacci cubes and Lucas cubes

Abstract

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