74 research outputs found
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
We present an efficient quantum algorithm for the exact evaluation of either
the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function
Z for a family of graphs related to irreducible cyclic codes. This problem is
related to the evaluation of the Jones and Tutte polynomials. We consider the
connection between the weight enumerator polynomial from coding theory and Z
and exploit the fact that there exists a quantum algorithm for efficiently
estimating Gauss sums in order to obtain the weight enumerator for a certain
class of linear codes. In this way we demonstrate that for a certain class of
sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon)
graphs, quantum computers provide a polynomial speed up in the difference
between the number of edges and vertices of the graph, and an exponential speed
up in q, over the best classical algorithms known to date
On the colored Tutte polynomial of a graph of bounded treewidth
AbstractWe observe that a formula given by Negami [Polynomial invariants of graphs, Trans. Amer. Math. Soc. 299 (1987) 601–622] for the Tutte polynomial of a k-sum of two graphs generalizes to a colored Tutte polynomial. Consequently, an algorithm of Andrzejak [An algorithm for the Tutte polynomials of graphs of bounded treewidth, Discrete Math. 190 (1998) 39–54] may be directly adapted to compute the colored Tutte polynomial of a graph of bounded treewidth in polynomial time. This result has also been proven by Makowsky [Colored Tutte polynomials and Kauffman brackets for graphs of bounded tree width, Discrete Appl. Math. 145 (2005) 276–290], using a different algorithm based on logical techniques
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
Evaluating a weighted graph polynomial for graphs of bounded tree-width
We show that for any there is a polynomial time algorithm to evaluate the weighted graph polynomial of any graph with tree-width at most at any point. For a graph with vertices, the algorithm requires arithmetical operations, where depends only on
Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
We define 2-decompositions of ribbon graphs, which generalise 2-sums and
tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial
of such a 2-decomposition, and derive the classical Brylawski formula for the
Tutte polynomial of a tensor product as a (very) special case. This study was
initially motivated from knot theory, and we include an application of our
formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric
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