Cube Polynomial of Fibonacci and Lucas Cubes

Abstract

The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k≥0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.