67 research outputs found
A weighted graph polynomial from chromatic invariants of knots
Motivated by the work of Chmutov, Duzhin and Lando on Vassiliev invariants, we define a polynomial on weighted graphs which contains as specialisations the weighted chromatic invariants but also contains many other classical invariants including the Tutte and matching polynomials. It also gives the symmetric function generalisation of the chromatic polynomial introduced by Stanley. We study its complexity and prove hardness results for very restricted classes of graphs
Monotone functions and maps
In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2
(2011)] we defined semi-monotone sets, as open bounded sets, definable in an
o-minimal structure over the reals, and having connected intersections with all
translated coordinate cones in R^n. In this paper we develop this theory
further by defining monotone functions and maps, and studying their fundamental
geometric properties. We prove several equivalent conditions for a bounded
continuous definable function or map to be monotone. We show that the class of
graphs of monotone maps is closed under intersections with affine coordinate
subspaces and projections to coordinate subspaces. We prove that the graph of a
monotone map is a topologically regular cell. These results generalize and
expand the corresponding results obtained in Basu et al. for semi-monotone
sets.Comment: 30 pages. Version 2 appeared in RACSAM. In version 3 Corollaries 1
and 2 were corrected. In version 4 Theorem 3 is correcte
Relations between M\"obius and coboundary polynomial
It is known that, in general, the coboundary polynomial and the M\"obius
polynomial of a matroid do not determine each other. Less is known about more
specific cases. In this paper, we will try to answer if it is possible that the
M\"obius polynomial of a matroid, together with the M\"obius polynomial of the
dual matroid, define the coboundary polynomial of the matroid. In some cases,
the answer is affirmative, and we will give two constructions to determine the
coboundary polynomial in these cases.Comment: 12 page
On the Potts model partition function in an external field
We study the partition function of Potts model in an external (magnetic)
field, and its connections with the zero-field Potts model partition function.
Using a deletion-contraction formulation for the partition function Z for this
model, we show that it can be expanded in terms of the zero-field partition
function. We also show that Z can be written as a sum over the spanning trees,
and the spanning forests, of a graph G. Our results extend to Z the well-known
spanning tree expansion for the zero-field partition function that arises
though its connections with the Tutte polynomial
On some algebraic identities and the exterior product of double forms
We use the exterior product of double forms to reformulate celebrated
classical results of linear algebra about matrices and bilinear forms namely
the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton
identities and Jacobi's formula for the determinant. This new formalism is then
used to naturally generalize the previous results to higher multilinear forms
namely to double forms.
In particular, we show that the Cayley-Hamilton theorem once applied to the
second fundamental form of a hypersurface of the Euclidean space is equivalent
to a linearized version of the Gauss-Bonnet theorem, and once its
generalization is applied to the Riemann curvature tensor (seen as a
double form) is an infinitisimal version of the general Gauss-Bonnet-Chern
theorem. In addition to that, the general Cayley-Hamilton theorems generate
several universal curvature identities. The generalization of the classical
Laplace expansion of the determinant to double forms is shown to lead to new
general Avez type formulas for all Gauss-Bonnet curvatures.Comment: 32 pages, in this new version we added: an introduction to the
exterior and composition products of double forms, a new section about
hyperdeterminants and hyperpfaffians and reference
Algorithms for Enumerating Circuits in Matroids
We present an incremental polynomial-time algorithm for enumerating all circuits of a matroid or, more generally, all minimal spanning sets for a flat. This result implies, in particular, that for a given infeasible system of linear equations, all its maximal feasible subsystems, as well as all minimal infeasible subsystems, can be enumerated in incremental polynomial time. We also show the NP-hardness of several related enumeration problems
Ideal hierarchical secret sharing schemes
Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.Peer ReviewedPostprint (author's final draft
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
Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips
We determine the general structure of the partition function of the -state
Potts model in an external magnetic field, for arbitrary ,
temperature variable , and magnetic field variable , on cyclic, M\"obius,
and free strip graphs of the square (sq), triangular (tri), and honeycomb
(hc) lattices with width and arbitrarily great length . For the
cyclic case we prove that the partition function has the form ,
where denotes the lattice type, are specified
polynomials of degree in , is the corresponding
transfer matrix, and () for ,
respectively. An analogous formula is given for M\"obius strips, while only
appears for free strips. We exhibit a method for
calculating for arbitrary and give illustrative
examples. Explicit results for arbitrary are presented for
with and . We find very simple formulas
for the determinant . We also give results for
self-dual cyclic strips of the square lattice.Comment: Reference added to a relevant paper by F. Y. W
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