The independence polynomial I(G, x) of a graph G is the polynomial in variable x in which the coefficient an on x n gives the number of independent subsets S ⊆ V (G) of vertices of G such that |S| = n. I(G, x) is unimodal if there is an index μ such that a0 ≤ a1 ≤ · · · ≤ aμ−1 ≤ aμ ≥ aμ+1 ≥ · · · ≥ ad−1 ≥ ad. While the independence polynomials of many families of graphs with highly regular stru...

A graph with at most two vertices of the same degree is called antiregular [25], maximally nonregular [32] or quasiperfect [2]. If sk is the number of independent sets of cardinality k in a graph G, then I(G;x) = s0 + s1x+ ...+ sαx α is the independence polynomial of G [10], where α = α(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence pol...

Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M . The ideas of rigidity independence and parallel independence, as well as Laman’s and Recski’...

Bayesian Knowledge Bases BKB are a rule based probabilistic model that extend Bayes Net works BN by allowing context sensitive independence and cycles in the directed graph BKBs have probabilistic semantics but lack independence semantics i e a graph based scheme determin ing what independence statements are sanctioned by the model Such a semantics is provided through generalized d separation b...

Bayesian Knowledge Bases (BKB) are a rule-based probabilistic model that extend Bayes Networks (BN), by allowing context-sensitive independence and cycles in the directed graph. BKBs have probabilistic semantics, but lack independence semantics, i.e., a graphbased scheme determining what independence statements are sanctioned by the model. Such a semantics is provided through generalized dsepar...

Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M . The ideas of rigidity independence and parallel independence, as well as Laman’s and Recski’...

In this paper we give a deterministic polynomial-time approximation algorithm for computing complex-valued evaluations of the Tutte polynomial and the independence polynomial on bounded degree graphs, as well for computing partition functions of complex-valued spin and edge-coloring models on bounded degree graphs. Our work builds on a recent line of work initiated by Barvinok [2, 3, 4, 5], whi...

In this talk, we will discuss some questions related to polynomial approximations of AC0. A classic result due to Tarui (1991) and Beigel, Reingold, and Spielman (1991), states that any AC0 circuit of size s and depth d has an -error probabilistic polynomial over the reals of degree at most (log(s/))ˆ{O(d)}. We will have a re-look at this construction and show how to improve the bound to (log s...

We construct a commutative algebra Az , generated by d algebraically independent q-difference operators acting on variables z1, z2, . . . , zd, which is diagonalized by the multivariable Askey-Wilson polynomials Pn(z) considered by Gasper and Rahman [6]. Iterating Sears’ 4φ3 transformation formula, we show that the polynomials Pn(z) possess a certain duality between z and n. Analytic continuati...

We consider multivariate polynomials with exponents that are themselves integer-valued multivariate polynomials, and we present algorithms to compute their GCD and factorization. The algorithms fall into two families: algebraic extension methods and projection methods. The first family of algorithms uses the algebraic independence of x, x, x 2 , x, etc, to solve related problems with more indet...