com2mac conference on combinatorial matrix...

POHANG UNIVERSITY OF SCIENCE AND TECHNOLOGY

Com2MaC CONFERENCE

ON COMBINATORIAL MATRIX THEORY

14-17 January, 2002

Organized and sponsored by the

Combinatorial and Computational Mathematics Center

Organizing Committee

Richard A. Brualdi (Co-chair) Han-Hyuk Cho Suk-Geun Hwang(Co-chair) Hyun Kwang Kim Arnold R. Krauter Sang-Gu Lee Bryan L. Shader Jia-Yu Shao

Com2MaC Conference on

Combinatorial Matrix Theory

Edited by

Richard A. Brualdi

Department of Mathematics

Univ. of Wisconsin-Madison

U.S.A.

Suk-Geun Hwang

Department of Mathematics

Kyungpook National Univ.

Korea

Supported by Com2MaC-KOSEF, Korea.

Jan. 14 - Jan. 17, 2002

Contents

Organization ii

I Program 1

II Abstracts 10

Invited Speakers 11

Registered Speakers 19

Index 30

z Maps around campus 31

i

Com2MaC Conference on Combinatorial Matrix Theory 2002.1.14-1.17

Com2MaC Conference on

Combinatorial Matrix Theory

CONFERENCE CHAIRS

Richard A. Brualdi, Univ. of Wisconsin-Madison, USASuk-Geun Hwang, Kyungpook National Univ., Korea

ORGANIZING COMMITTEE

Han-Hyuk Cho, Seoul Nat. Univ., KoreaHyunKwang Kim, POSTECH, Korea

Arnold Krauter, Univ. of Leoben, AustriaSang-Gu Lee, SungKyunKwan Univ., Korea

Bryan Shader, Univ. of Wyoming, USAJia-Yu Shao, Tongji University, P.R.China

INVITED SPEAKERS

Richard Brualdi(Plenary), Univ. of Wisconsin-Madison, U.S.A.Steve Kirkland(Plenary), Univ. of Regina, Canada

Arnold Kraeuter(Plenary), Univ. of Leoben, AustriaJian Shen(Plenary), Southwest Texas State Univ., USA

Bryan Shader(Plenary), Univ. of Wyoming, USAJia-yu Shao(Plenary), Tongji Univ., P.R.China

Miroslav Fiedler, Czech Academy of Sciences, Czech Rep.Willem Haemers, Katholieke Univ., Netherlands

Charles Johnson, College of William and Mary, USAQiao Li, Shanghai Jiao Tong Univ., ChinaZhongshan Li, Georgia State Univ. USA

Bolian Liu, South China Normal Univ., ChinaKazuyoshi Okubo, Hokkaido Univ. of Education, Japan

Chan Onn, National University of SingaporeJennifer Seberry, Univ. of Wollongong, Australia

Bit-Shun Tam, Tamkang University, TaiwanIan Wanless, Oxford Univ., U.K.

SPONSORS

Korea Science and Engineering FoundationMinistry of Science and Technology of Korea

PUBLICATIONS

Selected papers presented at this conference will be published in a special issue of the journal

Linear Algebra and Its Applications (LAA)Participants presenting papers are invited to submit the presented papers for publication in thisspecial issue of LAA to one of the special editors: Suk-Geun Hwang, Arnold Kraeuter, Bryan

Shader, and Jia-yu Shao. Papers will be refereed in accordance with the normal high standards ofLAA. April 30, 2002 is the deadline for papers to be submitted to one of the special editors.

Combinatorial and Computational Mathematics Center ii

Part I

Program

2

List of Participants

Ando, Tsuyoshi (Japan) Back, Ran (Korea) Beasley, LeRoy (USA) Bhowmik, Gautami (France) Brualdi, Richard (USA) Cha, Yong-Seok (Korea) Cheon, Gi-Sang (Korea) Cho, Han-Hyuk (Korea) Cho, Myeong-Sook (Korea) Choi, Ji-Young (USA) Choi, Kyung Sik (Korea) Chun, Hyunsuk (Korea) Fallat, Shaun (Canada) Fiedler, Miroslav (Czech Rep) Fonseca, Carlos M (Portugal) Furtado, Susana (Portugal) Gasparyan, Armen (Russia) Ha, Kil-Chan (Korea) Haemers, Willem (Netherlands) Hogben, Leslie (USA) Hwang, Geum Suk (Korea) Hwang, Suk-Geun (Korea) Jin, Min Ae (Korea) Johnson, Charles (USA) Kim, Duk-Sun (Korea) Kim, HyunKwang (Korea) Kim, Ik-Pyo (Korea) Kim, In-Jae (USA) Kim, Jeong Hwa(Korea) Kim, Jin-Soo (Korea) Kim, Se-Hoon (Korea) Kim, Suh-Ryung (Korea) Kimura, Yasunori (Japan) Kirkland, Steve (Canada) Koh, Young-Mee (Korea)

Kraeuter, Arnold (Austria) Lee, Eun-Young (Korea) Lee, Gwang-Yeon (Korea) Lee, Sang-Gu (Korea) Li, Jason Zhongshan (USA) Li, Qiao (China) Liu, Bolian (Chian) Maduka, Michael (Nigeria) McKay, Brendan (Australia) Nam, Yunsun (Korea) Neal, Cora (USA) Okubo, Kazuyoshi (Japan) Onn, Chan (Singapore) Park, Eun-Kyoung (Korea) Park, Jin-Woo (Korea) Park, Joo-Young (Korea) Pyo, Sung-Soo (Korea) Ree, Sang-Wook (Korea) Rho, Yumi (Korea) Rothblum, Uriel G. (Israel) Saiago, Carlos (Portugal) Sato, Kenzi (Japan) Seberry, Jennifer (Australia) Seo, Kyung Rok (Korea) Seol, Han-Guk (Korea) Shader, Bryan (USA) Shao, Jia-yu (China) Shchukin, Mikhail (Minsk) Shen, Jian (USA) Shin, Sun-Jeong (Korea) Siggers, Mark H. (USA) Song, Suk-Zun (Korea) Tam, Bit-Shun (Taiwan) Wanless, Ian (UK) Zhan, Xingzhi (Japan)

3

Time Table

Jan. 14, Monday 08:00-08:40 : Registration, 08:40-09:00 : Welcome by the Director

Parallel Session 1 (Room #1)

Jan.14(Mon) Jan.15(Tue) Jan.16(Wed) Jan.17(Thr)

09:00-10:00 R. Brualdi J. Shao A. Krauter B. Shader

10:05-10:45 K. Okubo B. McKay C. Onn B.-S. Tam

10:45-11:15 å Coffee Break

11:15-11:40 H.-G. Seol Beasley & Kirkland C. Neal G. Bhowmik

11:45-12:10 R. Baek S. Furtado G.-S. Cheon

12:15-12:40 S. Fallat S.-G. Lee

12:40-14:10 ð Lunch Break

14:10-15:10 J. Shen S. Kirkland

15:15-15:55 I. Wanless C. Johnson


16:25-16:50 H. Cho & S.-R. Kim U. Rothblum

16:55-17:20 S.-Z.Song S.Rhee &Y.Koh

Dinner at Faculty Lounge

Trip to

Kyoungju

Parallel Session 2 (Room #2)

Jan.14(Mon) Jan.15(Tue) Jan.16(Wed) Jan.17(Thr)

09:00-10:00 R. Brualdi J. Shao A. Krauter B. Shader

10:05-10:45 M. Fiedler B. Liu Q. Li W. Haemers


11:15-11:40 K. Sato X. Zhan T. Ando A. Gasparyan

11:45-12:10 Y. Rho G.-Y.Lee L. Hogben

12:15-12:40 C. Fonseca I. -J. Kim

12:40-14:10 ð Lunch Break

14:10-15:10 J. Shen S. Kirkland

15:15-15:55 J. Z. Li J. Seberry


16:25-16:50 S.-G. Hwang J.-Y. Choi

16:55-17:20 M. Maduka M. Shchukin

Dinner at Faculty Lounge

Trip to

Kyoungju

4

PROGRAM

Monday, January 14 08:00-08:40 Registration 08:40-09:00 Welcome by the Director (Room 1) MORNING SESSION

Room 1 09:00-10:00 Richard A. Brualdi (Univ. of Wisconsin-Madison, USA) Linear preservers and diagonal hypergraphs

Parallel Session I (Room 1)

10:05-10:45 Kazuyoshi Okubo (Hokkaido Univ. of Education, Japan) Rank reducing matrix norms 10:45-11:15 åCoffee Break 11:15-11:40 Han-Guk Seol (Sungkyunkwan Univ., Korea) Principally centrogonal matrices 11:45-12:10 Ran Back (Honam Univ, Korea)

An analysis of parallel algorithms of the eigenproblem for symmetric

Toeplitz matrices 12:15-12:40 Shaun Fallat (Univ. of Regina, Canada) On graphs with algebraic connectivity equal to minimum edge density

Parallel Session II (Room 2)

10:05-10:45 Miroslav Fiedler (Czech Acad. of Sciences, Czech Rep.) Basic matrices 10:45-11:15 åCoffee Break 11:15-11:40 Kenzi Sato (Tamagawa Univ., Japan) Free groups on spheres 11:45-12:10 Yumi Rho (Seoul Univ., Korea) On Kramer-Menser matrix partitioning conjecture

12:15-12:40 Carlos Martins da Fonseca (Univ. de Coimbra, Portugal) On the path polynomial of some matrices

5

AFTERNOON SESSION

Room1 14:10-15:10 Jian Shen (Southwest Texas State Univ., USA) Some problems on digraphs


15:15-15:55 Ian Wanless (Oxford Univ., U.K.) Permanents and rook polynomials 15:55-16:25 åCoffee Break 16:25-16:50 Han Hyuk Cho (Seoul National Univ., Korea) and Suh-Ryung Kim (Kyunghee Univ., Korea) Prime and semiprime matrices

16:55-17:20 Suk-Zun Song (Cheju Univ., Korea) The extremal values of permanent of (0,1)-matrices


15:15-15:55 Jason Zhongshan Li (Georgia State Univ., USA) Inertia sets of symmetric sign pattern matrices

15:55-16:25 åCoffee Break 16:25-16:50 Suk-Geun Hwang (Kyungpook Univ, Korea) Matrices determined by a linear recurrence relation among entries

16:55-17:20 Michael Maduka (Awka, Anambra State, Nigeria) TBA

6

Tuesday, January 15 MORNING SESSION Room1 09:00-10:00 Jia-Yu Shao (Tongji Univ., China) Matrices with special signed generalized inverses


10:05-10:45 Brendan McKay (Australian National Univ., Australia) Asymptotic enumeration of 0-1 matrices with prescribed line sums

10:45-11:15 åCoffee Break 11:15-11:40 LeRoy Beasley (Utah State Univ., USA) and Steve Kirkland(Univ. of Regina, Canada) k-primitive matrices 11:45-12:10 Susana Furtado (Univ. do Porto, Portugal) Spectral variation under congruence

12:15-12:40 Sang-Gu Lee (Sungkyunkwan Univ., Korea) JAVA matrix algorithms


10:05-10:45 Bolian Liu (South China Normal Univ., China) Generalized exponents of Boolean matrices

10:45-11:15 åCoffee Break 11:15-11:40 Xingzhi Zhan (Peking Univ., China and Tohoku Univ., Japan) On permutations of matrix entries

11:45-12:10 Gwang-Yeon Lee (Hanseo Univ., Korea) The linear algebra of the k-Fibonacci matrix

12:15-12:40 In-Jae Kim (Univ. of Wyoming, USA) The maximum spread over the class of symmetric (0,1)-matrices

7

AFTERNOON SESSION Room1 14:10-15:10 Steve Kirkland (Univ. of Regina, Canada) An approach to algebraic connectivity via nonnegative matrices


15:15-15:55 Charles Johnson (College of William and Mary, USA) A review of spectral theory for real symmetric matrices with a given graph

15:55-16:25 åCoffee Break 16:25-16:50 Uriel G. Rothblum (Technion, Israel), Partition polytopes

16:55-17:20 Sang-Wook Ree and Young-Mee Koh (Univ. of Suwon, Korea ) On hypertournament matrices


15:15-15:55 Jennifer Seberry (Univ. of Wollongong, Australia) The maximal determinant and subdeterminants of +,- 1 matrices

15:55-16:25 åCoffee Break 16:25-16:50 Ji-Young Choi (Iowa State Univ., USA) The P_0 matrix completion problem

16:55-17:20 Mikhail Shchukin (Belarusian State Univ., Minsk) The minimal number of idempotent generators for certain matrix algebras

äåDinner at Faculty Lounge

8

Wednesday, January 16

MORNING SESSION

Room1 09:00-10:00 Arnold Kraeuter (Univ. of Leoben, Austria) Bounds for the permanent of nonnegative matrices


10:05-10:45 Chan Onn (Singapore Nat. Univ., Singapore) Immanantal polynomials of laplacian matrices

10:45-11:15 åCoffee Break 11:15-11:40 Cora Neal (Weber State Univ., USA) 2-primitivity of tournaments

11:45-12:10 Gi-Sang Cheon (Daejin Univ., Korea) Sparsity of orthogonal matrices with both a row and a column of nonzero entries


10:05-10:45 Qiao Li (Shanghai Jiao Tong Univ., China) An approach to solving $A^k = J - I$

10:45-11:15 åCoffee Break 11:15-11:40 Tsuyoshi Ando (Hokusei Gakuen Univ., Japan) Maximal set of matrices with same inertia

11:45-12:10 Leslie Hogben (Iowa State Univ., USA) The matrix completion problems for pairs of related classes of matrices

Trip to Kyoungju in the afternoon

9

Thursday, January 17

MORNING SESSION

Room1 09:00-10:00 Bryan L. Shader (Univ. of Wyoming, USA) Exponents of tuples of nonnegative matrices


10:05-10:45 Bit-Shun Tam (Tamkang Univ., Taiwan) Graphs for cone-preserving maps revisited

10:45-11:15 åCoffee Break 11:15-11:40 Gautami Bhowmik (Univ. Lille I, France) Matrix Arithmetic and related questions


10:05-10:45 Willem Haemers (Katholieke Univ. Brabant, Netherlands) Which graphs are determined by their spectrum?

10:45-11:15 åCoffee Break 11:15-11:40 Armen Gasparyan (Program Systems Institute of RAS, Research

Centre of Multiprocessor Systems, Yaroslavl Region, Russia) Multidimensional matrix networks: A new approach to combinatorial problems

Part II

Abstracts

=======================================

Invited Speakers=======================================


Generalized Exponents of Boolean Matrices

Liu BolianDepartment of Mathematics

South China Normal University, P. R. [email protected]

In 1990 R.A. Brualdi and Bolian Liu (J.G.T 14(1990),483-499) introduced the concept of gener-alized exponent for primitive Boolean Marrices. In the study of generalized exponents, we focus onproblems of the following three aspacts. (1) To determine the maximums (namely, the exact upperbounds) of all kinds of generalized exponents. (2)To characterize the extreme martices completely;(3)To determine the sets of exponents (including the existence of gaps and their destributions). Inthe survey we will describe respectively the developments of these problems in recent years.

Linear Preservers and Diagonal Hypergraphs

Richard A. BrualdiDepartment of Mathematics

University of Wisconsin - [email protected]

Linear preservers of matrix functions, matrix properties, and matrix relations is a highly developedand active area of matrix theory. For certain special (combinatorial) linear preservers, the diagonalhypergraph of a matrix provides an alternative approach that does not rely on the classical approach.We give an introduction to results and problems in diagonal hypergraphs and relate this to (combi-natorial) linear preservers.

Combinatorial and Computational Mathematics Center 12


Immanantal Polynomials of Laplacian Matrices

Onn ChanDepartment of Mathematics

National University of Singapore, Republic of [email protected]

Joint work with Everton [email protected]

The immanent dλ(·) associated with the irreducible character χλ of the symmetric group Sn,indexed by the partition λ of n, acting on an n× n matrix A = [aij ] is defined by

dλ(A) =∑

σ∈Sn

χλ(σ)n∏

i=1

aiσ(i).

For a tree T on n vertices, let L(T ) denote its Laplacian matrix. Let x be an indeterminate variableand I be the n× n identity matrix. The immanantal polynomial of T corresponding to dλ is definedas

dλ(xI − L(T )) =n∑

k=0

(−1)kcλ,k(T )xn−k.

The coefficients cλ,k(T ) admit various algebraic and combinatorial interpretations for the tree T . Westudy the properties of cλ,k(T ) as well as upper and lower bounds on cλ,k(T ) and in particular showthat

cλ,k(S(n)) ≤ cλ,k(T ) ≤ cλ,k(P (n))

for all partitions λ and 0 ≤ k ≤ n, where S(n) and P (n) denote the star and the path on n verticesrespectively.

Basic matrices

Miroslav FiedlerDepartment of Mathematics

Czech Academy of Sciences, Czech [email protected]

For m × n matrices, structure means any non-void subset of M × N , M = {1, . . . ,m}, N ={1, . . . , n}. Given a matrix A (over a field) and a structure, the structure rank of A is the maximumrank of any submatrix of A all entries of whose are contained in the structure. We first surveyproperties of some structure ranks: subdiagonal rank, superdiagonal rank, etc.

Basic matrices are then defined as square irreducible matrices whose both subdiagonal and superdiag-onal ranks are one.

The class of n × n basic matrices contains tridiagonal matrices, inverses of nonsingular tridiagonalmatrices and is a proper extension of these for n > 2.

The inverse of a nonsingular basic matrix is again a basic matrix. We also assign to a basic matrixa lower and an upper envelope and define involutory relationships among lower and among upperenvelopes. These play a role for inverse basic matrices.

There is also a close relationship with the bidiagonal factorization. This approach has applications,e.g. for orthogonal and oscillatory matrices. In particular, basic oscillatory matrices have importantproperties and form, in a sense, the set of multiplicative generators of the class of oscillatory matrices.



Which graphs are determined by their spectrum?

Willem HaemersDept. of Econometrics and Operations Research

Tilburg University,Tilburg, The [email protected]

This question is old, but far from solved. It is conceivable that almost all graphs are determinedby the spectrum of its adjacency matrix (for short: DS), but it is also still possible that almost nographs are DS.

For non-DS graphs, several constructions are known. Schwenk proved (in 1974) that almost alltrees are not DS, and Godsil and McKay showed that the proportion of non-DS graphs on n verticesis at least c(n3)/(2n) for some constant c.

Most graphs that are known to be DS are either small, so that the property can be proved bycomplete enumeration, or they have a high degree of regularity. The latter examples are mainlydistance-regular graphs and line graphs of smaller DS graphs.

The aim of the talk is to survey the known DS and non-DS graphs.

A Review of Spectral Theory for Real Symmetric Matrices With a GivenGraph

Charles JohnsonDepartment of Mathematics

College of William and Mary, [email protected]

Let G be an undirected graph on n vertices and let S(G) denote the collection of all real symmetricmatrices whose graph is (precisely) G. This conveys no restriction upon the diagonal entries of thematrices. It is not surprising that G places some restrictions upon the spectral structure of thematrices in S(G), both the multiplicities of the (ordered) eigenvalues and the relative magnitudes ofthe entries of the eigenvectors (or combinatorial structure of the eigenspaces). A survey will be givenof recent work of this type.



An Approach to Algebraic Connectivity Via Nonnegative Matrices

Steve KirklandDepartment of Mathematics and Statistics

University of Regina, [email protected]

For a connected graph G, its Laplacian matrix L can be written as L=D-A, where A is the (0,1)adjacency matrix of G, and D is the diagonal matrix of vertex degrees. The algebraic connectivity of Gis the smallest positive eigenvalue of L, and as the name suggests, this eigenvalue and its correspondingeigenvectors seem to reflect some features of the graph. Not surprisingly, a good deal of the work onalgebraic connectivity relies on techniques from the theory of positive semidefinite matrices.

In this talk, we will discuss another approach to algebraic connectivity (and the associated eigen-vectors) that involves techniques from the theory of entrywise nonnegative matrices. This approachcan be particularly useful if the graph contains cutpoints, and by way of illustration, we will surveysome recent results that are derived through the nonnegative matrix approach. We will also showhow the technique can be extended to discuss the second smallest eigenvalue of any symmetric matrixwith nonpositive off-diagonal entries.

Bounds for the permanent of nonnegative matricesArnold R. Krauter

Department of Mathematics and Applied GeometryUniversity of Leoben

A-8700 Leoben, [email protected]

We present a review of recent developments on bounds for the permanent on nonnegative matrices.A part of the results being discussed originates in a collaboration with Suk-Geun Hwang and T. S.Michael.



An Approach to Solving Ak = J − I

Y. Wu and Q. Li*Dept. of Appl. Mathematics

Shanghai Jiao Tong Univ., P. R. [email protected]

We demonstrate an approach to verifying our conjecture on the complete solution to the matrixequation Ak = J − I, where k is a given positive integer, I is the identity matrix, J is the all onesmatrix, and A is an unknown (0, 1) matrix. To support our approach, we not only report what wehave obtained but also point out the gap between our current knowledge and the final settlement ofthe conjecture by posing several open problems remained to be answered along this approach.

Inertia Sets of Symmetric Sign Pattern Matrices

Frank J. Hall and Zhongshan (Jason) Li∗

Department of Mathematics and StatisticsGeorgia State University, U.S.A.

[email protected], [email protected]

A sign pattern matrix is a matrix whose entries are from the set {+,−, 0}. The symmetric signpattern matrices that require unique inertia have recently been characterized. The purpose of thispaper is to more generally investigate the inertia sets of symmetric sign pattern matrices. In particular,nonnegative tri-diagonal sign patterns and the square sign pattern with all + entries are examined.An algorithm is given for generating nonnegative real symmetric Toeplitz matrices with zero diagonalof orders n ≥ 3 which have exactly two negative eigenvalues. The inertia set of the square patternwith all + off-diagonal entries and zero diagonal entries is then analyzed. The types of inertias whichcan be in the inertia set of any sign pattern are also obtained in the paper. Specifically, certaincompatibility and consecutiveness properties are established. Finally, several sharp bounds for thesymmetric minimum rank of a tree sign pattern are obtained.

Rank Reducing Matrix Norms

K. OkuboMathematics Laboratory

Hokkaido University of Education-Sapporo, [email protected]

Joint work with H. J. Woerdeman

We consider approximation numbers for some norms on matrices, and look at the question whena closest rank ≤ p approximant can be chosen to reduce the rank of a matrix by p. If the latter isalways possible, we call the norm rank p reducing. It is easily seen that any unitarily invariant normis rank p reducing. We show that any absolute norm on Cn×m is rank n − 1 reducing and that thenumerical radius norm on Cn×n is rank n − 1 reducing as well. Non-examples and computations ofapproximation numbers are also presented.



The Maximal Determinant and Subdeterminants of +- 1 matrices

Jennifer SeberryDepartment of Mathematics

University of Wollongong, [email protected]

Hadamard matrices are orthogonal, square matrices which have entries +-1. They are know toonly exist for orders 1, 2 and 4m, m an integer. They are known to have maximal determinant if theyexist and they are conjectured to exist for all possible determinant. D-optimal designs are known tohave maximal determinant. They exist for orders 4m + 2 where 4m + 1 is the sum of two squares.We discuss what happens in other integers and how the subdeterminants of maximal determinantmatrices may be used.

Exponents of tuples of nonnegative matrices

Bryan ShaderDepartment of MathematicsUniversity of Wyoming, USA

[email protected] work with D.D. Olesky and P. van den Driessche

[email protected] and [email protected]

The notions of irreducibility, primitivity, and exponent of a nonnegative matrix are generalized tok-tuples of nonnegative matrices of the same order. It is shown that for each positive integer k, themaximum exponent of a primitive k-tuple of n by n nonnegative matrices is Θ(nk+1).

Matrices with special signed generalized inverses

Jia-yu ShaoDepartment of Mathematics

Tongji University, P. R. [email protected], [email protected]

A real matrix A is said to have a signed generalized inverse, if the sign pattern of its generalizedinverse A+ is uniquely determined by the sign pattern of A. A is said to have doubly signed generalizedinverse, if both A and A+ have a signed generalized inverse. A is said to have a nonnegative (or positive,or totally nonzero) signed generalized inverse, if A have a signed generalized inverse and A+ ≥ O (orA+ > O, or A+ contains no zero entries, respectively).

We obtain complete characterizations for the matrices to have doubly signed generalized inverses,to have nonnegative, or positive, or totally nonzero signed generalized inverses.



Some problems on digraphs

Jian ShenDepartment of Mathematics

Southwest Texas State University, [email protected], [email protected]

A primitive digraph is a strongly connected digraph for which the greatest common divisor of allcycle lengths is 1. We discuss the connections among various basic parameters in digraphs (especiallyin primitive digraphs): girth, diameter, vertex degrees, edge density, connectivity, exponent, etc.

Graphs for Cone-Preserving Maps Revisited

Bit-Shun TamDepartment of Mathematics

Tamkang University, Taipei, [email protected]

Joint work with Raphael Loewy

Let K be a proper (i.e., closed, pointed, full convex) cone in Rn and let A be an n-by-n realmatrix that satisfies AK ⊆ K. Let (E ,P(A,K)) denote the digraph with vertex set E consisting ofthe extreme rays of K such that (E1, E2) is an arc if and only if E2 ⊆ Φ(AE1), where Φ(S) denotesthe face of K generated by S. We show that the K-irreducibility or K-primitivity of A is completelydetermined by the digraph (E ,P(A, K)) together with a knowledge of when a given finite collection{E1, . . . , Ej} of extreme rays satisfies E1

∨ · · ·∨ Ej = K. We treat the problem of determining theexponent of a K-primitive matrix A. We also touch upon the question of when a given digraph Gallows (or requires) the existence of a (K-irreducible or K-primitive) matrix A ∈ π(K) such that(E ,P(A,K)) = G, with K being fixed or not fixed. Some open questions are also posed.

Permanents and Rook Polynomials

Ian WanlessChrist Church, St Aldates

Oxford University, United [email protected]

Rook polynomials count the number of ways in which non-attacking rooks can be placed on theblack squares of a chess board (of course, the black squares need not be arranged as per the traditionalboard). They are useful for finding the maximum permanent of certain classes of (0,1)-matrices. Iwill review these applications and their implications for perfect matchings in bipartite graphs andextensions to latin rectangles. I will also discuss a related open problem where we try to maximizethe number of 4-cycles in a class of bipartite graphs.


=======================================

Registered Speakers=======================================


==========================

Registered Speakers

==========================

Maximal set of matrices with same inertia

Tsuyoshi AndoFaculty of Economics

Hokusei Gakuen University, [email protected] or [email protected]

We give a characterization of a set of the form

{A ; A∗HA < H}

for some (hidden) definite or indefinite invertible matrix H. In a similar way we give a characterizationof a set of the form

{A ; HA + A∗H > 0}.

An Analysis of Parallel Algorithms of the Eigenproblem for symmetrictoeplitz matrices

Ran BaikDepartment of Computer and Applied Mathematics

Honam University , [email protected]

We propose the solution of symmetric toeplitz eigenvalue systems Tx = λx by the group homotopymethod. To solve this iterative method, we need the good choice of the initial matrices. In this case, wecan choose a circulant matrix C as an initial matrix. A convergence rate is governed by the eigenpairsof the initial matrices and it is fast if we choose a good circulant matrix which the eigenvalues of theinitial matrix is close to those of a given symmetric Toeplitz matrix and C minimizes ||T − C||1 and||T − C||∞. The method, when applied to a toeplitz eigenvalue problem, yields remarkable results,which are supported through numerical experiments.



k-primitive matrices

LeRoy BeasleyDepartment of Mathematics and Statistics

Utah State Univ., [email protected],[email protected]

Joint work with Steve Kirkland

Are all (0,1)-matrices which are primitive also expressible as a sum of k (0,1)-matrices (none ofthem the zero matrix) which yield a k-primitive matrix? Under what conditions is the answer yes?No? Partial answers will be given to these questions and examples to support negative answers.

Matrix Arithmetic and related questions

Gautami BhowmikUFR de MathematiquesUniversite Lille I, France

[email protected]

The study of arithmetic of integer matrices occurs naturally in the context of modular forms ofhigher degree. We shall discuss the basics of this arithmetic and show that the algebra of arithmeticfunctions is isomorphic to the Hall algebra. As a consequence we are able to derive information onthe number of subgroups of Abelian groups.

Prime and semiprime matrices

Han Hyuk Cho∗

Department of Mathematics EducationSeoul National University

[email protected] Kim

Department of MathematicsKyunghee University

[email protected]

A chain semiring is an algebraic system generalizing the Boolean algebra {0, 1} and the fuzzyalgebra [0, 1]. In this paper, as a generalization of prime Boolean matrices and prime fuzzy matrices,we study prime matrices and semiprime matrices over chain semirings using the relationship between(semi) prime matrices and their row spaces. We also study the factorization properties of the matriceswith full semiring rank, and give some combinatorial applications.



Sparsity of orthogonal matrices with both a row and a column of nonzeroentries

Gi-Sang CheonDepartment of Mathematics

Daejin Univ., [email protected]

It is known that the minimum number of nonzero entries in an m ×m orthogonal matrix whichhas a column of nonzeros is

(blg mc+ 3)m− 2blg mc+1.

In this paper, the sparsity of orthogonal matrices which have both a column and a row of nonzeros isstudied. It is conjectured that the minimum number of nonzero entries in such an m×m matrix is

(2

⌊lg

(23m

)⌋+ 3

)m− 5 · 2blg( 2

3 m)c + 3 + ε(m)

where ε(m) = m− 2blg(23 m)c+1 if m > 2blg(

23 m)c+1 and otherwise ε(m) = 0.

The P0-matrix Completion Problem

Ji Young ChoiDepartment of Mathematics

Iowa State Univ., [email protected]

In this paper we consider the P0-matrix completion problem. We establish that every asymmetricpartial P0-matrix has P0-completion. We also classify all 4 × 4 patterns that include all diagonalpositions as either having P0-completion or not having P0-completion. We show that any positionallysymmetric pattern whose graph is an n-cycle with n ≥ 5 has P0-completion.

On graphs with algebraic connectivity equal to minimum edge density

Shaun FallatDepartment of Mathematics and statistics

University of Regina, [email protected], [email protected]

Given an undirected graph G = (V,E) and a nonempty subset X ⊆ V , the edge density of Xis given by ρ(X) = |V ||EX |/|X||V \ X|, where EX is the set of all edges with one end in X andthe other end in V \X. It is known that the algebraic connectivity of G, denoted by a(G), satisfiesa(G) ≤ minX⊆V ρ(X). In this talk we study the graphs G for which equality holds in the aboveinequality. This is joint work with Dr. S. Pati and Prof. S. Kirkland.



On the path polynomial of some matrices

Carlos Martins da FonsecaDepartment of Mathematics

Universidade de Coimbra, [email protected]

The path polynomial, characteristic polynomial of the adjacency matrix of a path with k vertices,Pk, has been considered by several authors in the last decades. The objective of this talk is:

- describe completely the matrix Pk (Kn), where Kn is the complete graph on n vertices, usingsome important concepts of theory of circulant matrices.;

- make some considerations on path-positivity of several graphs;- evaluate the image of some matrices of graphs (like the Laplacian) by the path polynomial Pk.

Spectral Variation under Congruence

Susana FurtadoFaculdade de Economia da

Universidade do Porto, [email protected]

Joint work with Charles Johnson

The matrix B ∈ Mn is said to be congruent to A ∈ Mn if there is an invertible C ∈ Mn suchB = C∗AC. In this talk we describe the possible spectra among matrices congruent to a given A. It isimportant to distinguish singular A from nonsingular and, among nonsingular matrices, to distinguishwhere 0 lies relative to the field of values of A.

Multidimensional Matrix Networks: A new Approach to CombinatorialProblems

Armen GasparyanProgram Systems Institute of RAS

Research Centre of Multiprocessor SystemsYaroslavl Region, [email protected]

Most applications of matrix theory in combinatorics are of two types: first ones uses linear algebrato modeling combinatorial structures, and second ones – the multilinear algebra, that is determinants,permanents and other multilinear functions and forms.

In this paper is presented an extension of both paradigms proposing more general frameworkfor solving of combinatorial problems. Instead usual two- dimensional matrices we apply multiindexmatrices, and consequently instead usual determinants and permanents – multidimensional ones. Andin this way we obtain new unical possibilities for solving problems from enumerative and algebraiccombinatorics.



The paper consists of five parts. In first part are introduced the algebras of symmetric (multidi-mensional) matrices and corresponding algebras of symmetric functions with multiplications inducedfrom so called (λ, µ)-multiplications of matrices. This brings to a family of new operations and decom-position schemes for symmetric functions. Recall that the usual multiplication of symmetric functionscorresponds to tensorial multiplication, namely (0, 0)-multiplication. In second part the notion of ma-trix network is introduced. It is showed how can enumerative problems be bringed to computationalprocedures on specific matrix networks. As an important application we consider the general coloringenumeration problem with arbitrary system of conditions. Several powerful formulas are obtained.Third part is devoted especially to a famous problem of the counting of Latin rectangles. We introducethe notion of Latin configuration and consider general enumeration problem related to them. Thesolution is founded on a simple observation that the number of Latin configurations of given typeexactly equal to permanent of special multidimensional 0, 1-matrix, and on network decomposition ofthis matrix. Fourth part deals with relational networks, the matrix networks with multidimensional0, 1-matrix components and with boolean addition and multiplication. Matrices in that networks rep-resents relations, and boolean (λ, µ)-multiplications generalizes the join operators in usual relationalalgebra. We generalize this construction in several ways with transmission from boolean to multival-ued, fuzzy, stochastic and some other types of matrices and operations. Consequently we introduceand investigate multivalued- or fuzzy relational networks, stochastic- or probabilistic networks. Andin last part are proposed matrix models for processes in networks, in particular for logical inference,competition and gaming.

The Matrix Completion Problems for Pairs of Related Classes of Matrices

Leslie HogbenDepartment of Mathematics

Iowa State Univ., [email protected]

For a class X of real matrices, a list of positions in an n × n matrix (a pattern) is said to haveX-completion if every partial X-matrix that specifies exactly these positions can be completed to anX-matrix. If X and X0 are classes that satisfy the conditions (1) any partial X-matrix is a partial X0-matrix, (2) for any X0-matrix A and e > 0, A + e I is a X-matrix, and (3) for any partial X-matrix A,there exists d > 0 such that A - d is a partial X-matrix (where is the partial identity matrix specifyingthe same pattern as A). then any pattern that has X0-completion must also have X-completion.However, there are usually patterns that have X-completion that fail to have X0-completion. Thisresult applies to many pairs of subclasses of P- and P0-matrices defined by the same restrictionon entries, including the classes P/P0-matrices, (weakly) sign-symmetric P/P0-matrices, and non-negative P/P0-matrices. It also applies to other related pairs of subclasses of P0-matrices, suchas the pairs classes of P-/P0,1-matrices, (weakly) sign-symmetric P/P0,1-matrices and non-negativeP/P0,1-matrices. Furthermore, any pattern that has (weakly sign-symmetric, sign-symmetric, non-negative) P0-completion must also have (weakly sign-symmetric, sign-symmetric, non-negative) P0,1-completion, although these pairs of classes do not satisfy condition (3). Similarly, the class of inverseM-matrices and its topological closure do not satisfy condition (3), but the conclusion remains true,and the matrix completion problem for the topological closure of the class of inverse M-matrices issolved for patterns containing the diagonal.



Matrices determined by a linear recurrence relation among entries

Suk-Geun HwangDepartment of Mathematics Education

Kyungpook Natinal Univ., [email protected]

A matrix A = [ai,j ] is called a 7-matrix if its entries satisfy the recurrence (α)ai−1,j−1+(β)ai−1,j =ai,j for all i, j where alpha and beta are fixed numbers. A 7-matrix completely determined by itstop most row and right most column. In this paper we determine the structure of 7-matrices andinvestigate the sequences represented by columns of infinite 7-matrices.

The maximum spread over the class of symmetric (0, 1)-matrices

In-Jae KimDepartment of MathematicsUniversity of Wyoming, USA

[email protected]

The diameter of the set of all eigenvalues of a square, complex matrix is called its spread. Thistalk is concerned with the maximum spread over the collection of all n by n (0,1)-symmetric matrices.We reduce the problem to finding the maximum spread over a small subcollection of highly structuredmatrices, and then investigate properties of the spread over this subcollection.

The Linear Algebra of the k-Fibonacci matrix

Gwang-Yeon Lee∗

Department of MathematicsHanseo University, Korea

[email protected] Kim

Department of Mathematics, Sungkyunkwan UniversitySuwon 440-746, Korea

For a positive integer k ≥ 2, the k-Fibonacci sequence {g(k)n } is defined as: g

(k)1 = · · · = g

(k)k−2 = 0,

g(k)k−1 = g

(k)k = 1 and for n > k ≥ 2, g

(k)n = g

(k)n−1 + g

(k)n−2 + · · · + g

(k)n−k. The n × n k-Fibonacci matrix

GF (k)n = [fij ] is defined as: for fixed k ≥ 2,

fij ={

gi−j+1 i− j + 1 ≥ 0,0 i− j + 1 < 0,

where gn = g(k)n+k−2. If k = 2, then GF (2)n = Fn is the Fibonacci matrix. The properties of the

Fibonacci matrix is well-known.In this talk, we discuss the linear algebra of the k-Fibonacci matrix, and consider the symmetric

k-Fibonacci matrix.



JAVA matrix algorithms

Sang-Gu LeeDepartment of MathematicsSungkyunkwan Univ., Korea

[email protected]

We shall show multimedia web contents of Linear Algebra which was developed under KoreaResearch Foundation Project # 2000-048-D0001. This contains several JAVA applets, flash tools andanimations that enables the visualization of concepts in Lineal Algebra.

Asymptotic enumeration of 0-1 matrices with prescribed line sums

Brendan D. McKay∗ and Xiaoji WangDepartment of Mathematics

Australian National University, [email protected]

We consider 0-1 matrices of order n× n′ whose line (row and column) sums are specified. Equiv-alently, bicoloured simple graphs with specified degrees. The matrix is regular if all the line sums areequal.

We are interested in the asymptotic number of such matrices when n, n′ →∞ and the line sums arespecified as functions of n, n′. Let ∆ be the maximum line sum. Read (1958) solved the regular casewith ∆ = 3, and Everett and Stein (1971) extended this to arbitrary fixed ∆. The case of arbitraryline sums with bounded ∆ was solved by Bekessy, Bekessy and Komlos (1972), Bender (1974) andWormald (1978). When ∆ is allowed to increase with n, n′, the situation becomes more difficult.O’Neil (1969), Mineev and Pavlov (1976), and Bollobas and McKay (1986) did the regular case when∆ increased as a small power of log(n + n′). McKay (1984) extended this to the case ∆ = o(t1/4),where t is the total number of 1’s in the matrix; this includes the regular case with t = o(n1/3).

In this paper, we extend the theory to the case ∆ = o(t1/3).

2-Primitivity of Tournaments

Cora NealDepartment of Mathematics

Weber State Univ., [email protected]

It will be shown that all primitive tournaments can be non-trivially 2-colored so that the resulting2-colored tournament is 2-primitive. Also, asymptotically, almost all randomly 2-colored tournamentsare 2-primitive.



On k-Hypertournament Matrices

Sang-Wook Ree∗ and Youngmee KohDepartment of Mathematics

The University of Suwon, [email protected] and [email protected]

A k-hypertournament is a complete k-hypergraph with all k-edges endowed with orientations, i.e.,orderings of the vertices in the edges. The incidence matrix associated with a k-hypertournamentis called a k-hypertournament matrix, where each row stands for a vertex of the hypertournament.Some properties of the hypertournament matrices are investigated.

The sequences of the numbers of 1’s and −1’s of rows of a k-hypertournament matrix are respec-tively called the score sequence and the losing score sequence of the matrix and so of the correspondinghypertournament. A necessary and sufficient condition for a sequence to be the score sequence (re-spectively, the losing score sequence) of a k-hypertournament is proved.

We also find some conditions for the existence of k-hypertournament matrices with constant scoresequence, called regular k-hypertournament matrices.

On Kramer-Mesner Matrix Partitioning Conjecture

Yoomi RhoDepartment of MathematicsSeoul Natinal Univ., Korea

[email protected]

In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can bepartitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978,Kramer and Mesner conjectured that any mt×mt matrix with kt nonzero elements can be partitionedinto mn submatrices of order t of which at most k contain nonzero elements. In 1995, R. A. Brualdi et.al showed that this conjecture is true if m = 2, k ≤ 3 or k ≥ mn−2. They also found a counterexampleof this conjecture when m = 4, n = 4, k = 6 and t = 2. When t = 2, we show that this conjecture istrue if k = 4, k = 5 or k = mn− 3 by showing that it is true if m = 3.

Partition Polytopes

Uriel G. RothblumIndustrial Engineering and Management

Technion, [email protected]

We study polytopes that are the covex null of vectors associated with partitions of a set of realnumbers, where the vector associated with a partition are sums of the elements in each part. Theface structure of these polytopes is explored and the results are applied to optimization problems overpartitions.



Free Groups on Spheres

Kenzi SATODepartment of Mathematics, Faculty of Engineering

Tamagawa University, [email protected]

The famous following paradox was proved by using the axiom of choice:

The Hausdorff-Banach-Tarski Paradox If U and V are any bounded subsets of the 3-dimensionalEuclidean space R3, each having non-empty interior, then U and V can be both partitioned into thesame finite number of respectively congruent pieces.

The existence of a free subgroup of rank 2 of the special orthogonal group SO3(R) is useful toprove it. The problem of the existence of a free subgroup is generalized to:

For a non-empty set X and a group G acting on X, does there exist a free subgroup F of rank 2 ofG whose action on X is locally commutative, i.e., two elements of F which have a common fixed pointin X are commutative? Moreover, does there exist a free subgroup F of rank 2 of G which acts with-out non-trivial fixed points on X, i.e., no element of F distinct from the identity has a fixed point in X?

We consider the Hausdorff-Banach-Tarski Paradox for the n-dimensional rational Euclidean spaceQn when n ≥ 3, because it is shown without the axiom of choice unlike the paradox for the completeEuclidean space Rn (so the axiom of choice is not the essential reason of the Hausdorff-Banach-TarskiParadox). In order to prove the paradox for Qn, we want to show the following problems raised by J.Mycielski:

(A) Can we find a free subgroup F of rank 2 of the special orthogonal group with rational entriesSOn(Q) which acts without non-trivial fixed points on the unit sphere Sn−1 when n ≥ 4 is even?

(B’) For positive rational q, can we find a free subgroup F of rank 2 of SOn(Q) whose action onSn−1 is locally commutative and which acts without non-trivial fixed points on (

√qSn−1) ∩ Qn,

the rational sphere with radius√

q, when n ≥ 3 is odd?

We have already solved affirmatively them except the problem (A) for n ≡ 2 (mod 4) and theproblem (B’) for n ≡ 1 (mod 4) when q is square rational .

The minimal number of idempotent generators for certain matrix algebras

Mikhail ShchukinDepartment of Mathematics

Belarusian State University, [email protected]

Every homogeneous C∗-algebra corresponds to the algebraic fibre bundle. C∗-algebra is callednon-trivial if the corresponding algebraic fibre bundle is non-trivial. All C∗-algebras generated bythe idempotents that studied before corresponds to the trivial algebraic fibre bundles. I show thatnon-trivial C∗-algebras of any dimension n ≥ 2 can be generated by three idempotents. It is a minimalnumber of idempotent generators for such algebras.



Principally Centrogonal matrices

Han-Guk SeolDepartment of MathematicsSungkyunkwan Univ., Korea

[email protected] work with LeRoy Beasley and Sang-Gu Lee

A real nonsingular n × n matrix A = (aij) is called centrogonal if A−1 = (an+1−i,n+1−j); it iscalled principally centrogonal if all leading principal submatrices of A are centrogonal; and it is calledinverse principally centrogonal if A−1 is principally centrogonal. We give some results for the inverseprincipally centrogonal matrices.

Extremes of permanents of (0,1) matrices

Suk-Zun SongDepartment of MathematicsCheju National Univ., Korea

[email protected]

Let Mm,n(k) be the set of m by n matrices with k 0’s and mn − k 1’s. We shall determine themaximum and minimum values of permanent function over Mm,n(k), and the matrices which attainthe extreme values.

On permutations of matrix entries

Xingzhi ZhanDepartment of Mathematics

Peking Univ, now at Tohoku University, [email protected]

We determine those permutation operators that preserve eigenvalues, determinant, singular valuesor the set of positive semidefinite matrices. We also study the norm variations of a matrix underpermutations of its entries.


Index

Armen Gasparyan, 23Arnold R. Krauter, 15

Bit-Shun Tam, 18Brendan D. McKay, 26Bryan Shader, 17

Carlos Martins da Fonseca , 23Charles Johnson, 14Cora Neal, 26

Frank J. Hall, 16

Gautami Bhowmik, 21Gi-Sang Cheon, 22Gwang-Yeon Lee, 25

Han Hyuk Cho, 21Han-Guk Seol, 29

Ian Wanless, 18In-Jae Kim, 25

Jennifer Seberry, 17Ji Young Choi, 22Jia-yu Shao, 17Jian Shen, 18Jin-Soo Kim, 25

K. Okubo, 16Kenzi SATO , 28

LeRoy Beasley, 21Leslie Hogben, 24Liu Bolian, 12

Mikhail Shchukin, 28Miroslav Fiedler, 13

Onn Chan, 13

Q. Li, 16

Ran Baik, 20Richard A. Brualdi, 12

Sang-Gu Lee, 26Sang-Wook Ree, 27Shaun Fallat, 22Steve Kirkland, 15Suh-Ryung Kim, 21Suk-Geun Hwang, 25Suk-Zun Song, 29Susana Furtado, 23

Tsuyoshi Ando, 20

Uriel G. Rothblum, 27

Willem Haemers, 14

Xiaoji Wang, 26Xingzhi Zhan, 29

Y. Wu, 16Yoomi Rho, 27Youngmee Koh, 27

Zhongshan (Jason) Li, 16

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