The In–Out Degree Polynomial of the Directed Fibonacci Cubes

The in–out degree polynomial of a directed graph is a binary polynomial that combines the indegree polynomial and the outdegree polynomial. In this paper, we give the in–out degree polynomials and their generating function of the directed Fibonacci cubes. As a sequence, we obtain the indegree polynomials and the outdegree polynomials of the cubes. We also study some properties of the coefficients and roots of these univariate polynomials. Finally, we find a close relationship between the indegree polynomials of the cubes and the Morgan–Voyce polynomials, as well as the Fibonacci polynomials.