On Symmetry of Independence Polynomials

An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_{0}+s_{1}x+s_{2}x^{2}+...+s_{\alpha(G)}x^{\alpha(G)} is called the independence polynomial of G (Gutman and Harary, 1983). If $s_{j}=s_{\alpha-j}$, 0=< j =< alpha(G), then I(G;x) is called symmetric (or palindromic). It is known that the graph G*2K_{1} obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanovic, 1998). In this paper we show that for every graph G and for each non-negative integer k =< mu(G), one can build a graph H, such that: G is a subgraph of H, I(H;x) is symmetric, and I(G*2K_{1};x)=(1+x)^{k}*I(H;x).Comment: 16 pages, 13 figure

Path decompositions of chains and circuits

Expressions for the path polynomials (see Farrell [1]) of chains and circuits are derived. These polynomials are then used to deduce results about node disjoint path decompositions of chains and circuits. Some results are also given for decompositions in which specific paths must be used

The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovasz local lemma -- which provides a sufficient condition for this to occur -- corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer and explicitly by Dobrushin. We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity. To be published in J. Stat. Phy

Combinatorial Route to Algebra: The Art of Composition & Decomposition

We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.Comment: 20 pages, 6 figure