65,411 research outputs found
Uniform Mixing and Association Schemes
We consider continuous-time quantum walks on distance-regular graphs of small
diameter. Using results about the existence of complex Hadamard matrices in
association schemes, we determine which of these graphs have quantum walks that
admit uniform mixing.
First we apply a result due to Chan to show that the only strongly regular
graphs that admit instantaneous uniform mixing are the Paley graph of order
nine and certain graphs corresponding to regular symmetric Hadamard matrices
with constant diagonal. Next we prove that if uniform mixing occurs on a
bipartite graph X with n vertices, then n is divisible by four. We also prove
that if X is bipartite and regular, then n is the sum of two integer squares.
Our work on bipartite graphs implies that uniform mixing does not occur on
C_{2m} for m >= 3. Using a result of Haagerup, we show that uniform mixing does
not occur on C_p for any prime p such that p >= 5. In contrast to this result,
we see that epsilon-uniform mixing occurs on C_p for all primes p.Comment: 23 page
Classification of small class association schemes coming from certain combinatorial objects
We explore two- or three-class association schemes. We study aspects of the structure of the relation graphs in association schemes which are not easily revealed by their parameters and spectra. The purpose is to develop some combinatorial methods to characterize the graphs and classify the association schemes, and also to delve deeply into several specific classification problems. We work with several combinatorial objects, including strongly regular graphs, distance-regular graphs, the desarguesian complete set of mutually orthogonal Latin squares, orthogonal arrays, and symmetric Bush-type Hadamard matrices, all of which give rise to many small-class association schemes. We work within the framework of the theory of association schemes.;Our focus is placed on the search for all isomorphism classes of association schemes and characterization of small-class association schemes of specific order. In particular, we examine two-class association schemes (strongly regular graphs) of order 64 and their three-class fission schemes. After we collect \u27feasible\u27 parameter sets for the putative association schemes, we make an attempt to check the realization (existence) of the parameter sets and describe the structure of the schemes chiefly by investigating the structure of their relation graphs. In the course of this thesis, we find a new way to construct orthogonal arrays and investigate their implications for strongly regular graphs, symmetric Bush-type Hadamard matrices, and three-class association schemes. We obtain several results regarding the characterization and classification of two- or three-class association schemes of order 64
Pseudo-centrosymmetric matrices, with applications to counting perfect matchings
We consider square matrices A that commute with a fixed square matrix K, both
with entries in a field F not of characteristic 2. When K^2=I, Tao and Yasuda
defined A to be generalized centrosymmetric with respect to K. When K^2=-I, we
define A to be pseudo-centrosymmetric with respect to K; we show that the
determinant of every even-order pseudo-centrosymmetric matrix is the sum of two
squares over F, as long as -1 is not a square in F. When a
pseudo-centrosymmetric matrix A contains only integral entries and is
pseudo-centrosymmetric with respect to a matrix with rational entries, the
determinant of A is the sum of two integral squares. This result, when
specialized to when K is the even-order alternating exchange matrix, applies to
enumerative combinatorics. Using solely matrix-based methods, we reprove a weak
form of Jockusch's theorem for enumerating perfect matchings of 2-even
symmetric graphs. As a corollary, we reprove that the number of domino tilings
of regions known as Aztec diamonds and Aztec pillows is a sum of two integral
squares.Comment: v1: Preprint; 11 pages, 7 figures. v2: Preprint; 15 pages, 7 figures.
Reworked so that linear algebraic results are over a field not of
characteristic 2, not over the real numbers. Accepted, Linear Algebra and its
Application
Coherence for indexed symmetric monoidal categories
Indexed symmetric monoidal categories are an important refinement of
bicategories -- this structure underlies several familiar bicategories,
including the homotopy bicategory of parametrized spectra, and its equivariant
and fiberwise generalizations.
In this paper, we extend existing coherence theorems to the setting of
indexed symmetric monoidal categories. The most central theorem states that a
large family of operations on a bicategory defined from an indexed symmetric
monoidal category are all canonically isomorphic. As a part of this theorem, we
introduce a rigorous graphical calculus that specifies when two such operations
admit a canonical isomorphism.Comment: 100 pages, 64 figures, 13 table
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
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