115 research outputs found
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Probabilistic analysis of euclidean multi depot vehicle routing and related problems
We consider a generalization of the classical traveling salesman problem: the multi depot vehicle routing problem (MDVRP). Let be a set of depots and be sets customers in with the usual Euclidean metric. A multi depot vehicle routing tour is a set of disjoint cycles such that all customers are covered and each cycle contains exactly one depot. The goal is to find a tour of minimum length. denotes the length of an optimal MDVRP tour for depot set and customer set . It is evident that the asymptotic behavior of \L(D,P) for tending to infinity depends on the customer-depot ratio . We study three cases: , for a constant , and k=\Omega(n^{1+\ee}) for \ee>0. In the first two cases we show that divided by converges completely to a constant if the customers and depots are given by iid random variables. In the last case we prove that the expected tour length divided by and multiplied by converges to a constant if the customers and depots are given by iid random variables with uniform distribution
Synchronizing permutation groups and graph endomorphisms
The current thesis is focused on synchronizing permutation groups and on graph endo-
morphisms. Applying the implicit classification of rank 3 groups, we provide a bound
on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph
endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of
class r, establish their relation to mixed MDS codes, investigate G-decompositions of
(non)-synchronizing semigroups, and analyse the kernel graph construction used in the
theorem of Cameron and Kazanidis which identifies non-synchronizing transformations
with graph endomorphisms [20].
The contribution lies in the following points:
1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a
complete list of small non-synchronizing groups of permutation rank 3 is provided
(see Chapter 3).
2. The singular endomorphisms of the Hamming graph and some related graphs are
characterised (see Chapter 5).
3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper-
cuboids for small values are counted, and their correspondence to mixed MDS
codes is unveiled (see Chapter 6).
4. The research on normalizing groups from [3] is extended to semigroups of the
form , and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling
problems (see Chapter 7).
5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are
hulls and it is conjectured that a hull on n vertices has minimal generating set of at
most n generators (see Chapter 8)
Categorification and applications in topology and representation theory
International audienc
Threshold elements and the design of sequential switching networks
Includes bibliographies."AD 657370."[by] A.K. Susskind, D.R. Haring [and] C.L. Liu
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Systems of difference equations as a model for the Lorenz system
We consider systems of difference equations as a model for the Lorenz system of differential equations. Using the power series whose coefficients are the solutions of these systems, we define three real functions, that are approximation for the solutions of the Lorenz system
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