Independence roots and independence fractals of certain graphs

Abstract

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.

Graphs Whose Certain Polynomials Have Few Distinct Roots

“…Corollary 23 (see [31]). The independence roots of the family { } and { } are real and dense in (−∞, −1/4].…”

“…, ). For further studies on independence polynomial and independence root refer to [31][32][33]. The path 4 on 4 vertices, for example, has one independent set of cardinality 0 (the empty set), four independent sets of cardinality 1, and three independent sets of cardinality 2; its independence polynomial is then ( 4 , ) = 1 + 4 + 3 2 .…”

On the growth of deviations

“…Consider the f -polynomial and the h-polynomial of the independence complex of these graphs. The roots of f for C n and P n are determined explicitly in [1] and are respectively c (n) s = − 1 2(1 + cos( 2s−1 n π)) s = 1, 2, . .…”

Independence equivalence classes of cycles

“…In other words, the independence polynomial of a graph G is the product of the independence polynomials of the connected components of G. We thus try to factorise I(C n , x) over Z[x] to investigate properties of the factors. The roots of the independence polynomials of cyclic graphs have been completely determined by Alikhani and Peng [1].…”