14 research outputs found
Homomorphisms and polynomial invariants of graphs
This paper initiates a general study of the connection between graph homomorphisms and the Tutte
polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte
polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials.
As an application, we describe in terms of homomorphism counting some fundamental evaluations of the
Tutte polynomial in abelian groups and statistical physics. We conclude the paper by providing a
homomorphism view of the uniqueness conjectures formulated by Bollobás, Pebody and Riordan.Ministerio de Educación y Ciencia MTM2005-08441-C02-01Junta de AndalucÃa PAI-FQM-0164Junta de AndalucÃa P06-FQM-0164
Irreducibility of the Tutte polynomial of an embedded graph
We prove that the ribbon graph polynomial of a graph embedded in
an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable
Universal Tutte characters via combinatorial coalgebras
The Tutte polynomial is the most general invariant of matroids and graphs
that can be computed recursively by deleting and contracting edges. We
generalize this invariant to any class of combinatorial objects with deletion
and contraction operations, associating to each such class a universal Tutte
character by a functorial procedure. We show that these invariants satisfy a
universal property and convolution formulae similar to the Tutte polynomial.
With this machinery we recover classical invariants for delta-matroids, matroid
perspectives, relative and colored matroids, generalized permutohedra, and
arithmetic matroids, and produce some new convolution formulae. Our principal
tools are combinatorial coalgebras and their convolution algebras. Our results
generalize in an intrinsic way the recent results of
Krajewski--Moffatt--Tanasa.Comment: Accepted version, 51p
The Jones-Krushkal polynomial and minimal diagrams of surface links
We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in
thickened surfaces. It states that any reduced alternating diagram of a link in
a thickened surface has minimal crossing number, and any two reduced
alternating diagrams of the same link have the same writhe. This result is
proved more generally for link diagrams that are adequate, and the proof
involves a two-variable generalization of the Jones polynomial for surface
links defined by Krushkal. The main result is used to establish the first and
second Tait conjectures for links in thickened surfaces and for virtual links.Comment: 32 pages, 20 figures, and 1 tabl
Generation of Graph Classes with Efficient Isomorph Rejection
In this thesis, efficient isomorph-free generation of graph classes with the method of
generation by canonical construction path(GCCP) is discussed. The method GCCP
has been invented by McKay in the 1980s. It is a general method to recursively generate
combinatorial objects avoiding isomorphic copies. In the introduction chapter, the
method of GCCP is discussed and is compared to other well-known methods of generation.
The generation of the class of quartic graphs is used as an example to explain
this method. Quartic graphs are simple regular graphs of degree four. The programs,
we developed based on GCCP, generate quartic graphs with 18 vertices more than two
times as efficiently as the well-known software GENREG does.
This thesis also demonstrates how the class of principal graph pairs can be generated
exhaustively in an efficient way using the method of GCCP. The definition and
importance of principal graph pairs come from the theory of subfactors where each
subfactor can be modelled as a principal graph pair. The theory of subfactors has
applications in the theory of von Neumann algebras, operator algebras, quantum algebras
and Knot theory as well as in design of quantum computers. While it was
initially expected that the classification at index 3 + √5 would be very complicated,
using GCCP to exhaustively generate principal graph pairs was critical in completing
the classification of small index subfactors to index 5¼.
The other set of classes of graphs considered in this thesis contains graphs without
a given set of cycles. For a given set of graphs, H, the Turán Number of H, ex(n,H),
is defined to be the maximum number of edges in a graph on n vertices without a
subgraph isomorphic to any graph in H. Denote by EX(n,H), the set of all extremal
graphs with respect to n and H, i.e., graphs with n vertices, ex(n,H) edges and no
subgraph isomorphic to any graph in H. We consider this problem when H is a set of
cycles. New results for ex(n, C) and EX(n, C) are introduced using a set of algorithms
based on the method of GCCP. Let K be an arbitrary subset of {C3, C4, C5, . . . , C32}.
For given n and a set of cycles, C, these algorithms can be used to calculate ex(n, C)
and extremal graphs in Ex(n, C) by recursively extending smaller graphs without any
cycle in C where C = K or C = {C3, C5, C7, . . .} ᴜ K and n≤64. These results are
considerably in excess of the previous results of the many researchers who worked on
similar problems. In the last chapter, a new class of canonical relabellings for graphs, hierarchical
canonical labelling, is introduced in which if the vertices of a graph, G, is canonically
labelled by {1, . . . , n}, then G\{n} is also canonically labelled. An efficient hierarchical
canonical labelling is presented and the application of this labelling in generation
of combinatorial objects is discussed
Computational techniques in graph homology of the moduli space of curves
The object of this thesis is the automated computation of the rational (co)homology
of the moduli spaces of smooth marked Riemann surfaces Mg;n. This is achieved by
using a computer to generate a chain complex, known in advance to have the same
homology as Mg;n, and explicitly spell out the boundary operators in matrix form.
As an application, we compute the Betti numbers of some moduli spaces Mg;n.
Our original contribution is twofold. In Chapter 3, we develop algorithms for the
enumeration of fatgraphs and their automorphisms, and the computation of the
homology of the chain complex formed by fatgraphs of a given genus g and number
of boundary components n.
In Chapter 4, we describe a new practical parallel algorithm for performing Gaussian
elimination on arbitrary matrices with exact computations: projections indicate
that the size of the matrices involved in the Betti number computation can easily
exceed the computational power of a single computer, so it is necessary to distribute
the work over several processing units. Experimental results prove that our
algorithm is in practice faster than freely available exact linear algebra codes.
An effective implementation of the fatgraph algorithms presented here is available
at http://code.google.com/p/fatghol. It has so far been used to compute the Betti
numbers of Mg;n for (2g + n) 6 6.
The Gaussian elimination code is likewise publicly available as open-source software
from http://code.google.com/p/rheinfall