On the roots of independence polynomials of almost all very well-covered graphs

Abstract

If s k denotes the number of stable sets of cardinality k in graph G, and (G) is the size of a maximum stable set, then I (G; x) = (G)

The Roller-Coaster Conjecture Revisited

“…In other words, for 1-well-covered graphs, the domain of the Roller-Coaster Conjecture can be shortened to { 2α 3 , 2α It is easy to see that H•K 1 is very well-covered for every graph H, and some properties of I (H • K 1 ; x) are presented in [18]. Several findings concerning the palindromicity of I (H • Y ; x) are proved in [17, ?, 30].…”

“…• there exist graphs whose independence polynomials have all the roots real (for example, K 1,3 -free graphs [5], P n • K 1 for any n ≥ 1 [18]);…”

Maximum Modulus of Independence Roots of Graphs and Trees

“…This construction of attaching a leaf to every vertex in a graph G is know as the graph star operation, the resulting graph denoted G * . Levit and Mandrescu [19] proved a formula for i(G * , x) in terms of i(G, x) for all graphs G. Using Maple and nauty [21], we were able exploit this formula to verify that all well-covered trees on n ≤ 40 vertices have their independence roots contained in the unit disk! This makes it extremely tempting to conjecture that the independence roots of all well-covered trees are contained in the unit disk.…”

On the Stability of Independence Polynomials

“…We show stability for all independence polynomials of graphs with independence number at most three, but for larger independence number we show that the independence polynomials can have roots arbitrarily far to the right. We provide families of graphs whose independence polynomials are stable and ones that are not, utilizing various graph operations.where i k is the number of independent sets of size k in G. We call the roots of i(G, x) the independence roots of G.Research on the independence polynomial and in particular, the independence roots, has been very active (see, for example, [2,3,4,5,8,14] and [13] for an excellent survey) since it was first defined by Gutman and Harary in 1983 [11] (including recent connections, in the multivariate case, to the hard core model in statistical physics [15]). On the nature of these roots, Chudnovsky and Seymour [7] showed that the independence roots of clawfree graphs are all real, and Brown and Nowakowski [5] showed that with probability tending to 1, a graph will have a nonreal independence root.Asking when the independence roots are all real is a very natural question, but what about their location in the complex plane?…”

“…Research on the independence polynomial and in particular, the independence roots, has been very active (see, for example, [2,3,4,5,8,14] and [13] for an excellent survey) since it was first defined by Gutman and Harary in 1983 [11] (including recent connections, in the multivariate case, to the hard core model in statistical physics [15]). On the nature of these roots, Chudnovsky and Seymour [7] showed that the independence roots of clawfree graphs are all real, and Brown and Nowakowski [5] showed that with probability tending to 1, a graph will have a nonreal independence root.…”