WIENER INDEX AND HOSOYA POLYNOMIAL OF FIBONACCI AND LUCAS CUBES

Sandi Klavzar (1) , Michel Mollard (2)

1 FMF - Faculty of Mathematics and Physics [Ljubljana]
2 IF - Institut Fourier

In the language of mathematical chemistry, Fibonacci cubes can be defined as the resonance graphs of fibonacenes. Lucas cubes form a symmetrization of Fibonacci cubes and appear as resonance graphs of cyclic polyphenantrenes. In this paper it is proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonacci numbers which in turn yields a closed formula for the Wiener index of Fibonacci cubes. Asymptotic behavior of the average distance of Fibonacci cubes is obtained. The generating function of the sequence of ordered Hosoya polynomials of Fibonacci cubes is also deduced. Along the way, parallel results for Lucas cubes are given.

Mots clés: Hypercubes; Cube polynomials; Fibonacci cubes; Lucas cubes; Generating functions; Zeros of polynomials; Unimodal sequences

Domaines: Combinatoire [math.CO]

Sandi Klavzar, Michel Mollard. WIENER INDEX AND HOSOYA POLYNOMIAL OF FIBONACCI AND LUCAS CUBES. 2011. ⟨hal-00624188v2⟩