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

Curl-conforming hierarchical vector bases for triangles and tetrahedra

A new family of hierarchical vector bases is proposed for triangles and tetrahedra. These functions span the curl-conforming reduced-gradient spaces of Nédélec. The bases are constructed from orthogonal scalar polynomials to enhance their linear independence, which is a simpler process than an orthogonalization applied to the final vector functions. Specific functions are tabulated to order 6.5. Preliminary results confirm that the new bases produce reasonably well-conditioned matrice

BRS Cohomology of a Bilocal Model

We present a model in which a gauge symmetry of a field theory is intrinsic in the geometry of an extended space time itself. A consequence is that the dimension of our space time is restricted through the BRS cohomology. If the Hilbert space is a dense subspace of the space of all square integrable $C^{\infty}$ functions, the BRS cohomology classes are nontrivial only when the dimension is two or four.Comment: 15 pages, LaTeX. The original title, ``On the Dimension of the Space time'', is change

Symmetry algebra for the generic superintegrable system on the sphere

The goal of the present paper is to provide a detailed study of irreducible representations of the algebra generated by the symmetries of the generic quantum superintegrable system on the $d$-sphere. Appropriately normalized, the symmetry operators preserve the space of polynomials. Under mild conditions on the free parameters, maximal abelian subalgebras of the symmetry algebra, generated by Jucys-Murphy elements, have unique common eigenfunctions consisting of families of Jacobi polynomials in $d$ variables. We describe the action of the symmetries on the basis of Jacobi polynomials in terms of multivariable Racah operators, and combine this with different embeddings of symmetry algebras of lower dimensions to prove that the representations restricted on the space of polynomials of a fixed total degree are irreducible