On Classification and
Characterization of Association Schemes:
C1. (with
B. Xu) On wreath product of one-class association
schemes, (http://arxiv.org/abs/1008.2228)
C2. (with
G. Bhattacharyya and R. Tanaka) Terwilliger Algebras of Wreath
Products of One-Class Association Schemes, Journal of
Algebraic Combinatorics, 31 (2010), 455 -- 466.
C3. (with
S. Bang) On Generalized Semidirect Product of Association Schemes, Discrete
Mathematics, 303
(2005), 5 -- 16.
C4. (with
S. Bang & M. Hirasaka) Semidirect Products
of Association Schemes, Journal
of Algebraic Combinatorics, 22 No.1
(2005), 23-38.
There are many ways to
construct new association schemes from old ones. Association schemes can be
built up from `smaller' ones; two important constructive methods are the direct
product and wreath product. Another way to construct new association schemes
from old is by fusion and fission processes--the processes in which a new
association scheme is obtained by combining or splitting relations of the old
scheme in a certain way (cf. R3 below). In this paper, the semidirect
product operation is introduced as yet another way to construct new association
schemes from smaller ones. In his work developing the theory of association
schemes as a 'generalized' group theory, Zieschang
introduced the concept of the semidirect product as a
possible product operation of certain association schemes in 1996. In 2000, Muzychuk generalized the Zieschang’s
product operation slightly. However, both Zieschang
and Muzychuk’s operations are restricted to taking
the product of an association scheme with a ‘thin’ association scheme. In this paper we extend the semidirect product operation into the entire set of
association schemes. We then derive a way to decompose certain association
schemes into smaller association schemes. We also investigate to what extent
this product helps us to understand and characterize the structure of
association schemes. We give some examples to show that the semidirect
product produces many schemes that cannot be described
as neither the direct product nor the wreath product of smaller schemes.
C5. (with
S. Bang) Characterization of Maximal Rational Circulant Association Schemes, Codes
and Designs, (eds. K. Arasu and A. Seress) dedicated to Dijen K.
Ray-Chaudhuri’s 65th Birthday, deGuyter,
An association scheme is circulant if its relation matrix is equivalent to a circulant matrix. All fusion schemes of the regular group
scheme of a cyclic group are circulant. Every maximal
rational circulant association scheme arises as a
fusion scheme of the regular group scheme according to a certain standard
partition of the cyclic group. In this paper we give a complete description of
the structure of maximal rational circulant
association schemes in terms of the direct and wreath products of trivial association
schemes of primie order.
C6. (with K. See) Association
Schemes of Small Order, Journal of Statistical Planning and Inference,
73 (1998) Nos. 1/2, 225-271
The association schemes of
order up to 15 were classified mostly by the collaborative efforts of Nomiyama, Hirasaka, and Suga in Bannai's school in 1994 -
1998. (Although there had been some sporadic results obtained by others, their
work has been developed independently.) In this paper, we collect all
isomorphism classes of association schemes of order up to 15, and survey main
tools that are useful for dealing with the classification problem of
association schemes of small order. Part of the paper deals with the
construction and enumeration of association schemes via Schur
rings and two ways of tensoring association matrices,
and part deals with the fusion and fission relations of association schemes of
a given order by using the notion of the wreath product and direct product of
association schemes. The association schemes are presented in Hasse diagrams of partially ordered sets under the fusion
relations.
Note: Hanaki
and Miyamoto enumerated all the isomorphism classes of association schemes up
to order 19. (For order 16 & 17, see Kyushu J. Math. 52, (1998)
No.2, 383-396, and for order 18 & 19, Korean J. Computational & Applied
Math. 5, (1998) No.3, 543-552.) In October 1998, Miyamoto informed me
that they enumerated symmetric association schemes of order 23 and all
primitive association schemes of order up to 24. (The details are found at
"http://math.shinshu-u.ac.jp/~hanaki".)
On Fusion and Fission
of Association Schemes:
F1. Class 3 Association
Schemes Whose Symmetrization Have Two Classes,
Journal of Combinatorial Theory A 70 (1995) No.1, 1-29. [MR# 96b:05176;
Zbl.Math.842.05099]
F2. Commutative
Association Schemes Whose Symmetrizations Have Two
Classes, Journal of Algebraic Combinatorics
5 (1996), 47-55. [MR#96k:05213;
Zbl.Math.843.05103].
F3. Fusion Relations in Products of Association
Schemes, Graphs and Combinatorics. 18 (2002) 655-665.
F4. Fission Schemes of
Pseudo Cyclic Association Schemes (in preparation)
F5. (with L. K. Jorgensen, G. A. Jones, M. H. Klin) The Normally Regular Digraphs, Association Schemes and Related Combinatorial Structures (in preparation)
F6. (with F. Adams, A. Jendreau, O.Olmez) Construction of Directed Strongly Regular Graphs Using Regular
Tournaments (in preparation)
In these papers we
investigated fusion and fission relations in commutative association schemes in
a systematic way by studying their character tables, and tried to classify
small class of association schemes that have a specific fusion or fission
pattern. Also, we investigated fusion relations between the association schemes
obtained as the wreath product, direct product and semidirect
product of various association schemes. In this direction of research, it is
natural to study the structure of regular directed graphs appeared as the
relation graphs of the schemes.
On Character Tables of
Association Schemes:
CT1. (with H. Tanaka) Group-Case
Commutative Association Schemes and Their Character Tables Proceedings of Conference on Algebriac
Combinatorics dedicating Eiichi Bannai's
60th birthday, held in Sendai, Japan, June 2006.
(also
see http://arxiv.org/abs/0809.0748)
CT2. (with
CT3. (with
CT4. (with
CT5. (with
E. Bannai and
CT6. (with
K. W. Johnson and J. D. H. Smith) Characters of Finite Quasigroups
VI: Critical Examples and Doubletons, European Journal of Combinatorics, 11 (1990), 267-275.
[MR#91f:20079; Zbl.Math.704.20056]
CT7. (with
CT8. (with
CT9. (with
CT10. (with
CT11. (with
These papers not only provide
the character tables of corresponding association schemes but also reveal many
interesting relationships between the character theory of association schemes
and those of groups, Moufang loops and quasigroups. In particular, they show that the character
tables of many classical groups and many permutation groups acting on finite
geometries can be obtained in a systematic way from the character tables of
corresponding association schemes.
On Block Designs:
D1 (With J. Hegeman, J. Langford, G. Bhattacharyya, and J. Kim) Some Existence and Construction Results of Polygonal Designs: European Journal of Combinatorics, 29 (2008), 1396-1407.
D2. (With
K. See and J. Stufken) On a Class of
Partially Balanced Incomplete Block Designs with Applications in Survey Sampling,
Communications In Statistics Theory and Methods
1 (1997), 1-13. [MR#98f:62021]
D3. (with
K. Driessel, K. See and J. Stufken)
Polygonal Designs: Some Existence and Non-existence Results, Journal
of Statistical Planning and Inference, 77 (1999), No.1, 155-166.
D4. (with
A. Bailer, K.See and J. Stufken)
Relative Efficiencies of Sampling Plans for Selecting a Small Number of
Units from a Rectangular Region, Journal
of Statistical computation and Simulation, 66 No. 4 (2000), 273-294.
D5. (with
K. See) Spatially Constrained Sampling, In The Encyclopedia of Environmetrics (eds. A. El-Shaarawi and
We construct a class of
balanced incomplete block designs, called balanced sampling plans excluding
contiguous units as a continuation of the work initiated by Hedayat,
Rao and Stufken in sampling
survey. We study on the structure of related designs through a combinatorial
enumeration method. We also investigate the efficiency and effectiveness of the
balanced sampling designs comparing with other conventional sampling designs
for fixed finite populations.
On Graphs:
G1. (with
O. Olmez) Construction
of directed strongly regular graphs using finite incidence structures
(also
see http://arxiv.org/abs/1006.5395)
G2. (with
D. Eisenstat, J. Kim, D. Watson) Strongly
Regular Graphs with Parameters (64, 28, 12, 12) (in preparation)
G3. Products of Distance
Regular Graphs, Utilitas Mathematica 29 (1986), 173-175 [MR#87g:05209]
G4. (F5) (with
L. K. Jorgensen, G. A. Jones, M. H. Klin) The
Normally Regular Digraphs, Association Schemes and Related Combinatorial
Structures
(in preparation)
Miscellany:
M1. (with C. Godsil) Association Schemes, in: C. Colbourn
and J. Dinitz (Eds.). The CRC Handbook of COMBINATORIAL DESIGNS
(2nd. Ed.), CRC Press inc., Boca Raton, 2007, 325-330.
M2. (with
D. Choi) Spanning Time for Abstract Biological
Codes, Proceedings of Hungarian-Korean Combinatorics Workshop held in June 2001, at Renyi Institute of Mathematics in
M3. Posets
Related to Some Association Schemes (unpublished preprint)
M4. Commutative
Association Schemes and Related Algebra, Proceedings of the Fifth KIT
Mathematics Workshop, "Korea Institute of Technology, Korea, August
1990, 5, 143-173. [MR#92d:05184]
M5. Commutative
Association Schemes and Their Fusion and Fission Collected Papers Dedicated
to Professor Yeonsik Kim on the Occasion the 60th
Birthday, Ed. I. Jeong, et al., University
publishers, Korea, 1992, 113-130.
M6. The Terwilliger Algebras of Wreath Product Association Schemes
(in preparation)
M7. (with
K. See) An approximation with Multilinear Models
for
M8. (with
M. Maxwell) Characterization of distance-regular graphs obtained from almost
perfect nonlinear functions (in preparation)
M9. (with
J. Kim) Classification of small class association schemes of order 64 (in
preparation)
M10. Almost perfect nonlinear
functions and almost bent functions (in preparation)
M11. (with
Sang-Gu Lee and Jihwa
Noh) Educational policy and curriculum
of Korean school
mathematics in the late 19th and early 20th
century J. Korea Soc. Math. Ed. (Ser. E) Communications
of Mathematical Education, Vol. 23, No 4, (2009), 1093 --
1130.
Lecture Notes:
* Lecture notes for the
courses that I have taught at ISU
L1. (M605) Designs, Codes,
and Association Schemes
L2. (M606) Topics in Combinatorics: Finite Sets, Extremal
Set Systems, and Relations
L3. (452X) Discrete
Mathematics for Information Assurance
L4. (M504-505) Abstract
Algebra: Groups, Rings, and Fields.
L5. (M533) Introduction
to Cryptography