Independence roots and independence fractals of certain graphs

Abstract

The independence polynomial of a graph G is the polynomial ∑ikxk, where ik denote the number of independent sets of cardinality k in G. In this paper, we obtain the relationships between the independence polynomial of path Pn and cycle Cn with Jacobsthal polynomial. We find all roots of Jacobsthal polynomial. As a consequence, the roots of independence polynomial of the family {Pn} and {Cn} are real and dense in \((-\infty,-\frac{1}{4}]\). Also we investigate the independence fractals or independence attractors of paths, cycles, wheels and certain trees.