q-counting hypercubes in Lucas cubes

Abstract

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