On the Roots of Expected Independence Polynomials

Abstract

The independence polynomial of a (finite) graph is the generating function for the number of independent sets of each cardinality. Assuming that each possible edge of a complete graph of order n is independently operational with probability p, we consider the expected independence polynomial. We show here that for all fixed p∈(0,1), the expected independence polynomials of complete graphs have all real, simple roots.

On a sequence of sparse binomial-type polynomials

“…Recently the authors (see [1,2]) have defined and used these polynomials in the following graph theoretical setting. An independent set of vertices of a (finite simple) graph is a subset of the vertices of the graph, no two of which are joined by an edge.…”

“…Then the expected number of independent sets of a graph of order n is given by f n (q). In [1,2] the authors study, among other things, the growth and asymptotic behavior of f n (q) for fixed real q with 0 < q < 1.…”

On a conjecture on sparse binomial-type polynomials by Brown, Dilcher and Manna

“…1) was recently studied in a number of papers by the authors Brown, Dilcher and Manna. Originally, this sequence of polynomials arises from the question on the expected number of independent sets of vertices of finite simple graphs, see [3] for more details. Moreover, those polynomials may also be considered to be interesting because of their close connection with the Jacobi Theta functions investigated in [4].…”

Derivatives and Real Roots of Graph Polynomials

“…Real roots of other graph polynomials have also been extensively studied, such as edgecover polynomial [2], the expected independence polynomial [7], domination polynomial [6], sigma-polynomial [67], chromatic polynomial [14,37,66], Wiener polynomial [12], flow polynomial [37], Tutte polynomial [16,64], etc. For more results on the roots of graph polynomials, we refer to [13,31,32,52,56,57,65].…”