8,009 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
Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
Inspired by some intriguing examples, we study uniform association schemes
and uniform coherent configurations, including cometric Q-antipodal association
schemes. After a review of imprimitivity, we show that an imprimitive
association scheme is uniform if and only if it is dismantlable, and we cast
these schemes in the broader context of certain --- uniform --- coherent
configurations. We also give a third characterization of uniform schemes in
terms of the Krein parameters, and derive information on the primitive
idempotents of such a scheme. In the second half of the paper, we apply these
results to cometric association schemes. We show that each such scheme is
uniform if and only if it is Q-antipodal, and derive results on the parameters
of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We
revisit the correspondence between uniform indecomposable three-class schemes
and linked systems of symmetric designs, and show that these are cometric
Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class
schemes in terms of only a few parameters, and show that any strongly regular
graph with a ("non-exceptional") strongly regular decomposition gives rise to
such a scheme. Hemisystems in generalized quadrangles provide interesting
examples of such decompositions. We finish with a short discussion of
five-class schemes as well as a list of all feasible parameter sets for
cometric Q-antipodal four-class schemes with at most six fibres and fibre size
at most 2000, and describe the known examples. Most of these examples are
related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions,
April 201
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Some Implications on Amorphic Association Schemes
AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition
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