3,884 research outputs found
Parity of Sets of Mutually Orthogonal Latin Squares
Every Latin square has three attributes that can be even or odd, but any two
of these attributes determines the third. Hence the parity of a Latin square
has an information content of 2 bits. We extend the definition of parity from
Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the
corresponding orthogonal arrays (OA). Suppose the parity of an
has an information content of bits. We show that
. For the case corresponding to projective
planes we prove a tighter bound, namely when
is odd and when is even. Using the
existence of MOLS with subMOLS, we prove that if
then for all sufficiently large .
Let the ensemble of an be the set of Latin squares derived by
interpreting any three columns of the OA as a Latin square. We demonstrate many
restrictions on the number of Latin squares of each parity that the ensemble of
an can contain. These restrictions depend on and
give some insight as to why it is harder to build projective planes of order than for . For example, we prove that when it is impossible to build an for which all
Latin squares in the ensemble are isotopic (equivalent to each other up to
permutation of the rows, columns and symbols)
Extrema of graph eigenvalues
In 1993 Hong asked what are the best bounds on the 'th largest eigenvalue
of a graph of order . This challenging question has
never been tackled for any . In the present paper tight bounds are
obtained for all and even tighter bounds are obtained for the 'th
largest singular value
Some of these bounds are based on Taylor's strongly regular graphs, and other
on a method of Kharaghani for constructing Hadamard matrices. The same kind of
constructions are applied to other open problems, like Nordhaus-Gaddum problems
of the kind: How large can be
These constructions are successful also in another open question: How large
can the Ky Fan norm be
Ky Fan norms of graphs generalize the concept of graph energy, so this question
generalizes the problem for maximum energy graphs.
In the final section, several results and problems are restated for
-matrices, which seem to provide a more natural ground for such
research than graphs.
Many of the results in the paper are paired with open questions and problems
for further study.Comment: 32 page
Generalizations of Choi's Orthogonal Latin Squares and Their Magic Squares
Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler.
He introduced a pair of orthogonal Latin squares of order 9 in his book.
Interestingly, his two orthogonal non-diagonal Latin squares produce a magic
square of order 9, whose theoretical reason was not studied. There have been a
few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei
Lih's construction of Choi's Latin squares of order 9 based on two
orthogonal Latin squares. In this paper, we give a new generalization of Choi's
orthogonal Latin squares of order 9 to orthogonal Latin squares of size
using the Kronecker product including Lih's construction. We find a geometric
description of Chois' orthogonal Latin squares of order 9 using the dihedral
group . We also give a new way to construct magic squares from two
orthogonal non-diagonal Latin square, which explains why Choi's Latin squares
produce a magic square of order 9.Comment: 18 pages revised slightly from Dec. 5, 2018 versio
E7 groups from octonionic magic square
In this paper we continue our program, started in [2], of building up
explicit generalized Euler angle parameterizations for all exceptional compact
Lie groups. Here we solve the problem for E7, by first providing explicit
matrix realizations of the Tits construction of a Magic Square product between
the exceptional octonionic algebra J and the quaternionic algebra H, both in
the adjoint and the 56 dimensional representations. Then, we provide the Euler
parametrization of E7 starting from its maximal subgroup U=(E6 x U(1))/Z3.
Next, we give the constructions for all the other maximal compact subgroups.Comment: 23 pages, added sections with new construction
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
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