349 research outputs found
Which Digraphs with Ring Structure are Essentially Cyclic?
We say that a digraph is essentially cyclic if its Laplacian spectrum is not
completely real. The essential cyclicity implies the presence of directed
cycles, but not vice versa. The problem of characterizing essential cyclicity
in terms of graph topology is difficult and yet unsolved. Its solution is
important for some applications of graph theory, including that in
decentralized control. In the present paper, this problem is solved with
respect to the class of digraphs with ring structure, which models some typical
communication networks. It is shown that the digraphs in this class are
essentially cyclic, except for certain specified digraphs. The main technical
tool we employ is the Chebyshev polynomials of the second kind. A by-product of
this study is a theorem on the zeros of polynomials that differ by one from the
products of Chebyshev polynomials of the second kind. We also consider the
problem of essential cyclicity for weighted digraphs and enumerate the spanning
trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for
publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00
Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking
We propose a new camera-based method of robot identification, tracking and
orientation estimation. The system utilises coloured lights mounted in a circle
around each robot to create unique colour sequences that are observed by a
camera. The number of robots that can be uniquely identified is limited by the
number of colours available, , the number of lights on each robot, , and
the number of consecutive lights the camera can see, . For a given set of
parameters, we would like to maximise the number of robots that we can use. We
model this as a combinatorial problem and show that it is equivalent to finding
the maximum number of disjoint -cycles in the de Bruijn graph
.
We provide several existence results that give the maximum number of cycles
in in various cases. For example, we give an optimal
solution when . Another construction yields many cycles in larger
de Bruijn graphs using cycles from smaller de Bruijn graphs: if
can be partitioned into -cycles, then
can be partitioned into -cycles for any divisor of
. The methods used are based on finite field algebra and the combinatorics
of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied
Mathematic
Eulerian digraphs and toric Calabi-Yau varieties
We investigate the structure of a simple class of affine toric Calabi-Yau
varieties that are defined from quiver representations based on finite eulerian
directed graphs (digraphs). The vanishing first Chern class of these varieties
just follows from the characterisation of eulerian digraphs as being connected
with all vertices balanced. Some structure theory is used to show how any
eulerian digraph can be generated by iterating combinations of just a few
canonical graph-theoretic moves. We describe the effect of each of these moves
on the lattice polytopes which encode the toric Calabi-Yau varieties and
illustrate the construction in several examples. We comment on physical
applications of the construction in the context of moduli spaces for
superconformal gauged linear sigma models.Comment: 27 pages, 8 figure
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
A problem on partial sums in abelian groups
In this paper we propose a conjecture concerning partial sums of an arbitrary
finite subset of an abelian group, that naturally arises investigating simple
Heffter systems. Then, we show its connection with related open problems and we
present some results about the validity of these conjectures
Regular configurations and TBR graphs
PhD 2009 QMThis thesis consists of two parts: The first one is concerned with the
theory and applications of regular configurations; the second one is devoted
to TBR graphs.
In the first part, a new approach is proposed to study regular configurations,
an extremal arrangement of necklaces formed by a given number
of red beads and black beads. We first show that this concept is closely related
to several other concepts studied in the literature, such as balanced
words, maximally even sets, and the ground states in the Kawasaki-Ising
model. Then we apply regular configurations to solve the (vertex) cycle
packing problem for shift digraphs, a family of Cayley digraphs.
TBR is one of widely used tree rearrangement operationes, and plays
an important role in heuristic algorithms for phylogenetic tree reconstruction.
In the second part of this thesis we study various properties
of TBR graphs, a family of graphs associated with the TBR operation.
To investigate the degree distribution of the TBR graphs, we also study
-index, a concept introduced to measure the shape of trees. As an interesting
by-product, we obtain a structural characterization of good trees,
a well-known family of trees that generalizes the complete binary trees
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