q-cube enumerator polynomial of Fibonacci cubes

• Published in: Discrete Applied Mathematics Vol. 226; pp. 127 - 137
• Main Authors: Saygı, Elif, Eğecioğlu, Ömer
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 31.07.2017; Elsevier BV
• Subjects: [formula omitted]-analogue; Bivariate analysis; Combinatorics; Cube enumerator polynomial; Cubes; Fibonacci cube; Fibonacci number; Fibonacci numbers; Graph theory; Hypercube; Hypercubes; Integrals; Numbers; Polynomials; Cube enumerator polynomial; q-analogue; Fibonacci cube; Fibonacci number; Hypercube
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2017.04.026

We consider a q-analogue of the cube polynomial of Fibonacci cubes. These bivariate polynomials satisfy a recurrence relation similar to the standard one. They refine the count of the number of hypercubes of a given dimension in Fibonacci cubes by keeping track of the distances of the hypercubes to the all 0 vertex. For q=1, they specialize to the standard cube polynomials. We also investigate the divisibility properties of the q-analogues and show that the quotient polynomials for the appropriate indices have nonnegative integral polynomials in q as coefficients. These results have many corollaries which include expressions involving the q-analogues of the Fibonacci numbers themselves and their convolutions as they relate to hypercubes in Fibonacci cubes. Many of our developments can be viewed as refinements of enumerative results given by Klavžar and Mollard in (2012).