Fractional strong matching preclusion for two variants of hypercubes

Let F be a subset of edges and vertices of a graph G. If G-F has no fractional perfect matching, then F is a fractional strong matching preclusion set of G. The fractional strong matching preclusion number is the cardinality of a minimum fractional strong matching preclusion set. In this paper, we mainly study the fractional strong matching preclusion problem for two variants of hypercubes, the multiply twisted cube and the locally twisted cube, which are two of the most popular interconnection networks. In addition, we classify all the optimal fractional strong matching preclusion set of each

Matrix Models on Large Graphs

We consider the spherical limit of multi-matrix models on regular target graphs, for instance single or multiple Potts models, or lattices of arbitrary dimension. We show, to all orders in the low temperature expansion, that when the degree of the target graph $\Delta\to\infty$, the free energy becomes independent of the target graph, up to simple transformations of the matter coupling constant. Furthermore, this universal free energy contains contributions only from those surfaces which are made up of ``baby universes'' glued together into trees, all non-universal and non-tree contributions being suppressed by inverse powers of $\Delta$. Each order of the free energy is put into a simple, algebraic form.Comment: 19pp. (uses harvmac and epsf), PUPT-139

Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov-Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful.Comment: 66 page

Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary TT-deformation of HOMFLY

We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of 2 steps: first, with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2^m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T-deformation of HOMFLY polynomial -- because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices. The second step is a certain minimization of residues of this new polynomial with respect to T+1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube -- this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov's construction at N=2. This second step is still somewhat sophisticated -- though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies.Comment: 146 pages; some points clarified, some typos correcte