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Let G = (V, E) be a simple graph. A set S ⊆ V is a dominating set of G, if every vertex in V \S is adjacent to at least one vertex in S. Let C i n be the family of dominating sets of a cycle C n with cardinality i, and let d(C n , i) = |C i n |. In this paper, we construct C i n , and obtain a recursive formula for d(C n , i). Using this recursive formula, we consider the polynomial D(C n , x) = n i=⌈ n 3 ⌉ d(C n , i)x i , which we call domination polynomial of cycles and obtain some properties of this polynomial.

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The independence polynomial of a graph G is the polynomial i k x k , where i k denote the number of independent sets of cardinality k in G. In this paper, we obtain the relationships between the independence polynomial of path P n and cycle C n with Jacobsthal polynomial. We find all roots of Jacobsthal polynomial. As a consequence, the roots of independence polynomial of the family {P n } and {C n } are real and dense in (−∞, − 1 4 ]. Also we investigate the independence fractals or independence attractors of paths, cycles, wheels and certain trees.