Abstract
An independent set in a graph G is a set of pairwise non-adjacent
vertices, and the independence number α(G) is the cardinality of a maximum stable set in G. The independence polynomial of G is
I(G;x)=s<sub>0</sub>+s<sub>1</sub>x+s<sub>2</sub>x<sup>2</sup>+...+s<sub>α</sub>x<sup>α</sup>, α=α(G),
where s<sub>k</sub> is the number of independent sets of size k in G (I. Gutman and F. Harary, 1983). The Fibonacci index Fib(G) of a graph G is the number of all its independent sets, i.e., Fib(G)=I( G;1). Tight lower and upper bounds for Fibonacci index are known for general graphs, connected or not, and the corresponding extremal graphs are characterized [6], [15].
In this paper, we give explicit formulae for independence polynomials of these extremal graphs, which we further use to analyze some of their properties (unimodality, log-concavity).