3,351 research outputs found
Symmetric Graphs and their Quotients
In this expository paper we describe a group theoretic characterization of
arc-transitive graphs and their quotients. When passing from a symmetric graph
to its quotient, much information is lost, but some of this information may be
recovered from a certain combinatorial design on the blocks, as well as a
bipartite graph between the blocks. We address the "extention problem" which
asks, when is this additional information sufficient to recover the original
graph?Comment: This is my honours thesis from 2005, its a bit of a mes
Quantum Walks on Embeddings
We introduce a new type of discrete quantum walks, called vertex-face walks,
based on orientable embeddings. We first establish a spectral correspondence
between the transition matrix and the vertex-face incidence structure.
Using the incidence graph, we derive a formula for the principal logarithm of
, and find conditions for its underlying digraph to be an oriented graph.
In particular, we show this happens if the vertex-face incidence structure
forms a partial geometric design. We also explore properties of vertex-face
walks on the covers of a graph. Finally, we study a non-classical behavior of
vertex-face walks
Classification of quasi-symmetric 2-(64,24,46) designs of Blokhuis-Haemers type
This paper completes the classification of quasi-symmetric 2-
designs of Blokhuis-Haemers type supported by the dual code of the
binary linear code spanned by the lines of initiated in
\cite{bgr-vdt}. It is shown that contains exactly 30,264
nonisomorphic quasi-symmetric 2- designs obtainable from maximal
arcs in via the Blokhuis-Haemers construction. The related strongly
regular graphs are also discussed.Comment: 11 page
Efficient Continuous-Duty Bitter-Type Electromagnets for Cold Atom Experiments
We present the design, construction and characterization of Bitter-type
electromagnets which can generate high magnetic fields under continuous
operation with efficient heat removal for cold atom experiments. The
electromagnets are constructed from a stack of alternating layers consisting of
copper arcs and insulating polyester spacers. Efficient cooling of the copper
is achieved via parallel rectangular water cooling channels between copper
layers with low resistance to flow; a high ratio of the water-cooled surface
area to the volume of copper ensures a short length scale ~1 mm to extract
dissipated heat. High copper fraction per layer ensures high magnetic field
generated per unit energy dissipated. The ensemble is highly scalable and
compressed to create a watertight seal without epoxy. From our measurements, a
peak field of 770 G is generated 14 mm away from a single electromagnet with a
current of 400 A and a total power dissipation of 1.6 kW. With cooling water
flowing at 3.8 l/min, the coil temperature only increases by 7 degrees Celsius
under continuous operation.Comment: 5 pages, 5 figure
Experimental Results of the Search for Unitals in Projective Planes of Order 25
In this paper we present the results from a program developed by the author
that finds the unitals of the known 193 projective planes of order 25.. There
are several planes for which we have not found any unital. One or more than one
unitals have been found for most of the planes. The found unitals for a given
plane are nonisomorphic each other. There are a few unitals isomorphic to a
unital of another plane. A t - (v; k; {\lambda}) design D is a set X of points
together with a family B of k-subsets of X called blocks with the property that
every t points are contained in exactly {\lambda} blocks. The design with t = 2
is called a block-design. The block-design is symmetric if the role of the
points and blocks can be changed and the resulting confguration is still a
block-design. A projective plane of order n is a symmetric 2-design with v = n2
+ n + 1, k = n + 1, {\lambda} = 1. The blocks of such a design are called
lines. A unital in a projective plane of order n = q2 is a set U of q3 + 1
points that meet every line in one or q + 1 points. In the case projective
planes of order n = 25 we have: q = 5, the projective plane is 2 - (651; 26; 1)
design, the unital is a subset of q3 + 1 = 53+ 1 = 126 points and every line
meets 1 or 6 points from the subse
A class of symmetric graphs with 2-arc-transitive quotients
Let be a finite X-symmetric graph with a nontrivial X-invariant
partition on such that is a
connected (X,2)-arc-transitive graph and is not a multicover of
. This article aims to give a characterization of
for the case where for
and . This investigation
requires a study on (X,2)-arc-transitive graphs of valency 4 or 7. We give a
characterization of tetravalent (X,2)-arc-transitive graphs at first; and as a
byproduct, we prove that every tetravalent (X,2)-transitive graph is either the
complete graph on 5 vertices or a near n-gonal graph for some . Then we
show that a heptavalent -arc-transitive graph can occur as
if and only if
for .Comment: 22 page
Symmetric graphs with complete quotients
Let be a -symmetric graph with vertex set . We suppose that
admits a -partition , with parts of
size , and that the quotient graph induced on is a complete
graph of order . Then, for each pair of distinct suffices , the
graph induced on the union is bipartite with each vertex of
valency or (a constant). When , it was shown earlier how a
flag-transitive -design induced on a part can sometimes be
used to classify possible triples . Here we extend
these ideas to and prove that, if the group induced by on a part
is -transitive and the "blocks" of have size less than ,
then either (i) , or (ii) the triple is known
explicitly.Comment: The first version of this manuscript dates from 2000. It was uploaded
to the arXiv since several people wished to have a copy. This new version is
updated with a literature review up to 2017. It is submitted for publication
and is currently under review (September 2017
Supersaturation of : from Zarankiewicz towards Erd\H{o}s-Simonovits-Sidorenko
For a positive integer , a graph and a bipartite graph let denote the number of copies of in , and let
denote the minimum number of copies of in all graphs
with edges. The study of such a function is the
subject of theorems of supersaturated graphs and closely related to the
Sidorenko-Erd\H{o}s-Simonovits conjecture as well. In the present paper we
investigate the case when and in particular the quadrilateral
graph case. For , we obtain exact results if and the corresponding
Zarankiewicz number differ by at most , by a finite geometric construction
of almost difference sets. if and the corresponding
Zarankiewicz number differs by we prove asymptotically sharp
results. We also study stability questions and point out the connections to
covering and packing block designs
Hypergraph encoding set systems and their linear representations
We study -designs of parameters over finite fields as
group divisible designs and set systems admitting a transitive action of a
linear group encoded in an hypergraph whose vertex set of size is
partitioned into sets of size in such a way that every -subset is
contained in at least subsets of . We relate the problem to the
representation theory of the general linear group \GL(n,\mathbb{F}_{q}) and
the constructions of AG codes over finite fields
Combinatorial -designs from special polynomials
Combinatorial -designs have nice applications in coding theory, finite
geometries and several engineering areas. There are two major methods of
constructing -designs. One of them is via group actions of certain
permutation groups which are -transitive or -homogeneous on some point
set. The other is a coding-theoretical one. The objectives of this paper are to
introduce two constructions of -designs with special polynomials over finite
fields GF, and obtain -designs and -designs with interesting
parameters. A type of d-polynomials is defined and used to construct
-designs. Under the framework of the first construction, it is shown that
every o-polynomial over GF gives a -design, and every o-monomial over
GF yields a -design. Under the second construction, every
-polynomial gives a -design. Some open problems and conjectures are also
presented in this paper
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