q-counting hypercubes in Lucas cubes
TURKISH JOURNAL OF MATHEMATICS, vol.42, no.1, pp.190-203, 2018 (SCI-Expanded, Scopus, TRDizin)
- Publication Type: Article / Article
- Volume: 42 Issue: 1
- Publication Date: 2018
- Doi Number: 10.3906/mat-1605-2
- Journal Name: TURKISH JOURNAL OF MATHEMATICS
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
- Page Numbers: pp.190-203
- Open Archive Collection: AVESIS Open Access Collection
- Hacettepe University Affiliated: Yes
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 KlavZar 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.