The independence polynomial of a graph G is the generating function I(G, x) = ∑ k≥0 ikx k, where ik is the number of independent sets of cardinality k in G. We show that the problem of evaluating the independence polynomial of a graph at any fixed non-zero number is intractable, evenwhen restricted to circulants. We provide a formula for the independence polynomial of a certain family of circul...

The independence polynomial i(G, x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing...

We generalize Gel’fond’s criterion of algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of C, instead of just one point (one extension deals with a subgroup of C).

Causal independence models offer a high level starting point for the design of Bayesian networks but are not maximally exploited as their behaviour is often unclear. One approach is to employ qualitative probabilistic network theory in order to derive a qualitative characterisation of causal independence models. In this paper we exploit polynomial forms of Boolean functions to systematically an...

We show that for n ≥ 3 there is an action of the braid group Bn on the determinantal ideals of a certain n × n symmetric matrix with algebraically independent entries off the diagonal and 2s on the diagonal. We show how this action gives rise to an action of Bn on certain compact subspaces of some Euclidean spaces of dimension n 2 . These subspaces are real semi-algebraic varieties and include ...

A graph G is well-covered if every maximal independent set has the same cardinality. Let sk denote the number of independent sets of cardinality k, and define the independence polynomial of G to be S(G, z) = ∑ skz k. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by zc wh...