Building Graphs Whose Independence Polynomials Have Only Real Roots

Abstract

A stable set in a graph G is a set of pairwise non-adjacent vertices, and the stability number α(G) is the maximum size of a stable set in G. The independence polynomial of G is

$$I(G; x) = s_{0}+s_{1}x+s_{2}x^{2}+\cdots+s_{\alpha}x^{\alpha},\alpha=\alpha(G),$$

where s _k equals the number of stable sets of cardinality k in G (Gutman and Harary [11]).

Unlike the matching polynomial, the independence polynomial of a graph can have non-real roots. It is known that the polynomial I(G; x) has only real roots whenever (a) α(G) = 2 (Brown et al. [4]), (b) G is claw-free (Chudnowsky and Symour [6]). Brown et al. [3] proved that given a well-covered graph G, one can define a well-covered graph H such that G is a subgraph of H, α(G) = α(H), and I(H; x) has all its roots simple and real.

In this paper, we show that starting from a graph G whose I(G; x) has only real roots, one can build an infinite family of graphs, some being well-covered, whose independence polynomials have only real roots (and, sometimes, are also palindromic).