16,425 research outputs found

    Relative Tutte polynomials of tensor products of colored graphs

    Full text link
    The tensor product (G1,G2)(G_1,G_2) of a graph G1G_1 and a pointed graph G2G_2 (containing one distinguished edge) is obtained by identifying each edge of G1G_1 with the distinguished edge of a separate copy of G2G_2, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollob\'as and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion-contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot

    Basis Criteria for Generalized Spline Modules via Determinant

    Full text link
    Given a graph whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling by the elements of R such that the difference of the labels on adjacent vertices lies in the ideal associated to the edge. The set of generalized splines has a ring and an R-module structure. We study the module structure of generalized splines where the base ring is a greatest common divisor domain. We give basis criteria for generalized splines on cycles, diamond graphs and trees by using determinantal techniques. In the last section of the paper, we define a graded module structure for generalized splines and give some applications of the basis criteria for cycles, diamond graphs and trees.Comment: 20 pages, 10 figure

    Binomial edge ideals and rational normal scrolls

    Full text link
    Let XX be the Hankel matrix of size 2Γ—n2\times n and let GG be a closed graph on the vertex set [n].[n]. We study the binomial ideal IGβŠ‚K[x1,…,xn+1]I_G\subset K[x_1,\ldots,x_{n+1}] which is generated by all the 22-minors of XX which correspond to the edges of G.G. We show that IGI_G is Cohen-Macaulay. We find the minimal primes of IGI_G and show that IGI_G is a set theoretical complete intersection. Moreover, a sharp upper bound for the regularity of IGI_G is given
    • …
    corecore