The independence polynomial has been widely studied in algebraic graph theory, in statistical physics, and in algorithms for counting and sampling problems. Seminal results of Weitz (2006) and Sly (2010) have shown that in bounded-degree graphs the independence polynomial can be eXciently approximated if the argument is positive and below a certain threshold, whereas above that threshold the po...

The independence polynomial has been widely studied in algebraic graph theory, in statistical physics, and in algorithms for counting and sampling problems. Seminal results of Weitz (2006) and Sly (2010) have shown that in bounded-degree graphs the independence polynomial can be efficiently approximated if the argument is positive and below a certain threshold, whereas above that threshold the ...

In this paper we define ?-independent (a weak-version of independence), Kronecker and dissociate sets on hypergroups and study their properties and relationships among them and some other thin sets such as independent and Sidon sets. These sets have the lacunarity or thinness property and are very useful indeed. For example Varopoulos used the Kronecker sets to prove the Malliavin theorem. In t...

One can define the independence polynomial of a graph G as follows. Let ik(G) denote the number of independent sets of size k of G, where i0(G) = 1. Then the independence polynomial of G is

We prove that for any set E ⊆ Z with upper Banach density d(E) > 0, the set “of cubic configurations” in E is large in the following sense: for any k ∈ N and any ε > 0, the set {(n1, . . . , nk) ∈ Z k : d( ⋂ e1,...,ek∈{0,1} (E − (e1n1 + · · ·+ eknk))) > d (E) k − ε} is an AVIP0set. We then generalize this result to the case “of polynomial cubic configurations” e1p1(n) + · · · + ekpk(n) where th...

Proof. Let f1, . . . , fm be generators of A(X) over A(Y ). Then the fi also generate K(X) over K(Y ), so we can reorder indices such that f1, . . . , fr are algebraically independent over K(Y ). Let R = A(Y )[f1, . . . , fr] ⊆ A(X). Since the fi are algebraically independent over K(Y ) they are algebraically independent over A(Y ), so R is isomorphic to an r-variable polynomial ring, which is ...

Let K/Q be a finite Galois extension with the Galois group G, and let χ be a nontrivial irreducible character ofG. Artin’s conjecture predicts that the L-function L(s, χ,K/Q) is holomorphic in the whole complex plane ([1], P. 105). Let χ1, . . . , χr be the irreducible nontrivial characters of G. The corresponding L-functions L(s, χ1), . . . , L(s, χr) are algebraically independent over C ([2],...

The independence polynomial of a graph G is the polynomial ΣA x |A|, summed over all independent subsets A ⊆ V (G). We prove that if G is clawfree, then all the roots of its independence polynomial are real. This extends a theorem of Heilmann and Lieb [12], answering a question posed by Hamidoune [11] and Stanley [15].

This paper offers a generalization of the independence polynomial, the co-k-plex polynomial. The resulting family of polynomials carries combinatorial information on a class of independence systems defined over the vertex set of a finite graph. Specifically, we offer a recursion formula and examples of the co-2-plex polynomial on certain graphs. In addition, we characterize the class of graphs ...