$q$-counting hypercubes in Lucas cubes

Abstract

Lucas and Fibonacci cubes are special subgraphs of the binary hypercubes that have been proposed as models of interconnection networks. Since these families are closely related to hypercubes, it is natural to consider the nature of the hypercubes they contain. Here we study a generalization of the enumerator polynomial of the hypercubes in Lucas cubes, which $q$-counts them by their distance to the all 0 vertex. Thus, our bivariate polynomials refine the count of the number of hypercubes of a given dimension in Lucas cubes and for $q=1$ they specialize to the cube polynomials of Klavžar and Mollard. We obtain many properties of these polynomials as well as the $q$-cube polynomials of Fibonacci cubes themselves. These new properties include divisibility, positivity, and functional identities for both families.