An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. If sk is the number of independent sets of cardinality k in G, then I(G;x) = s0 + s1x+ s2x 2 + ...+ sαx α, α = α (G) , is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If sα−i = f (i) · si holds for every i ∈ {0, 1, ..., ⌊α/2⌋}, t...

Let f1, . . . , fr denote a system of polynomials in the polynomial ring P = k[x1, . . . , xd] such that 0 = (0, . . . , 0) ∈ V (f1, . . . , fr) ⊂ AK . Suppose that fR is an xR-primary ideal, where R = P(x). Then the Hilbert-Samuel multiplicity e0(f ;R) provides a certain information about the local structure of the affine variety V = V (f1, . . . , fr) as considered in Bézout’s theorem and rel...

Recent work has detailed the conditions under which univariate Laurent polynomials have functional decompositions. This paper presents algorithms to compute such univariate Laurent polynomial decompositions efficiently and gives their multivariate generalization. One application of functional decomposition of Laurent polynomials is the functional decomposition of so-called “symbolic polynomials...

Matchings in graphs correspond to independent sets in the corresponding line graphs. Line graphs are an important subclass of claw-free graphs. Hence studying independence polynomials of claw-free graphs is a natural extension of studying matching polynomials of graphs. We extend a result of Bayati et.al. showing a fully polynomial time approximation scheme (FPTAS) for computing the independenc...

We consider the problem of approximating the independence number and the chromatic number of k-uniform hypergraphs on n vertices. For xed integers k 2, we obtain for both problems that one can achieve in polynomial time approximation ratios of at most O(n=(log 1) n)2). This extends results of Boppana and Halld orsson [5] who showed for the graph case that an approximation ratio of O(n=(logn)) c...

An independent set in a graph G is a set of pairwise non-adjacent vertices, and the independence number α(G) is the cardinality of a maximum independent set. The independence polynomial of G is I(G; x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α(G), where sk equals the number of independent sets of size k in G (Gutman and Harary, 1983). If si = sα−i, 0 ≤ i ≤ α/2 , then I(G; x) is called palindromic. I...

A stable set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is I(G;x) = s0+s1x+s2x 2 +...+sαx α , where α = α(G) is the cardinality of a maximum stable of G, while sk equals the number of stable sets of size k in G (Gutman and Harary, 1983). Hamidoune, 1990, showed that for every claw-free graph G (i.e., a graph having no induced subgraph isomorphic to...

We describe an inductive means of constructing infinite families of graphs, every one of whose members G has independence polynomial I(G; x) having only real zeros. Consequently, such independence polynomials are logarithmically concave and unimodal.

In this paper an upper bound for the number of algebraically independent Poincaré-Liapunov constants in a certain basis for planar polynomial differential systems is given. Finally, it is conjectured that an upper bound for the number of functionally independent Poincaré-Liapunov quantities would be m + 3m− 7 where m is the degree of the polynomial differential system. Moreover, the computation...