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Elif Saygi, Omer Egecioglu
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We consider a q-analogue of the cube polynomial of Fibonacci graphs. 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 graphs 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 graphs. Many of our developments can be viewed as refinements of enumerative results given by Klavzar and Mollard.