16,425 research outputs found
Relative Tutte polynomials of tensor products of colored graphs
The tensor product of a graph and a pointed graph
(containing one distinguished edge) is obtained by identifying each edge of
with the distinguished edge of a separate copy of , 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
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
Let be the Hankel matrix of size and let be a closed
graph on the vertex set We study the binomial ideal which is generated by all the -minors of which
correspond to the edges of We show that is Cohen-Macaulay. We find
the minimal primes of and show that is a set theoretical complete
intersection. Moreover, a sharp upper bound for the regularity of is
given
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