44 research outputs found
Four-class Skew-symmetric Association Schemes
An association scheme is called skew-symmetric if it has no symmetric
adjacency relations other than the diagonal one. In this paper, we study
4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of
skew-symmetric amorphous association schemes, submitted for publication], we
discovered that their character tables fall into three types. We now determine
their intersection matrices. We then determine the character tables and
intersection numbers for 4-class skew-symmetric pseudocyclic association
schemes, the only known examples of which are cyclotomic schemes. As a result,
we answer a question raised by S. Y. Song [Commutative association schemes
whose symmetrizations have two classes, J. Algebraic Combin. 5(1) 47-55, 1996].
We characterize and classify 4-class imprimitive skew-symmetric association
schemes. We also prove that no 2-class Johnson scheme can admit a 4-class
skew-symmetric fission scheme. Based on three types of character tables above,
a short list of feasible parameters is generated.Comment: 12 page
Association schemes from the action of fixing a nonsingular conic in PG(2,q)
The group has an embedding into such that it acts as
the group fixing a nonsingular conic in . This action affords a
coherent configuration on the set of non-tangent lines of the
conic. We show that the relations can be described by using the cross-ratio.
Our results imply that the restrictions and to the sets
of secant lines and to the set of exterior lines,
respectively, are both association schemes; moreover, we show that the elliptic
scheme is pseudocyclic.
We further show that the coherent configuration with even allow
certain fusions. These provide a 4-class fusion of the hyperbolic scheme
, and 3-class fusions and 2-class fusions (strongly regular graphs)
of both schemes and $R_{-}(q^2). The fusion results for the
hyperbolic case are known, but our approach here as well as our results in the
elliptic case are new.Comment: 33 page
Average mixing of continuous quantum walks
If is a graph with adjacency matrix , then we define to be the
operator . The Schur (or entrywise) product is a
doubly stochastic matrix and, because of work related to quantum computing, we
are concerned the \textsl{average mixing matrix}. This can be defined as the
limit of C^{-1} \int_0^C H(t)\circ H(-t)\dt as . We establish
some of the basic properties of this matrix, showing that it is positive
semidefinite and that its entries are always rational. We find that for paths
and cycles this matrix takes on a surprisingly simple form, thus for the path
it is a linear combination of , (the all-ones matrix), and a permutation
matrix.Comment: 20 pages, minor fixes, added section on discrete walks; fixed typo
Regular graphs with four eigenvalues
We study the connected regular graphs with four distinct eigenvalues. Properties and feasibility conditions of the eigenvalues are found. Several examples, constructions and characterizations are given, as well as some uniqueness and nonexistence results.Graphs;Eigenvalues;mathematics
Studies on non-amorphous association schemes and spin models
Tohoku University宗政昭弘課