September 15-18, 2011

2011 Workshop on Algebraic Combinatorics 2011代数组合论国际学术会议

Organizers

Eiichi Bannai Suogang GaoJun MaKaishun WangYaokun WuXiao-Dong Zhang

Shanghai Jiao Tong University Hebei Normal University Shanghai Jiao Tong University Beijing Normal University Shanghai Jiao Tong University Shanghai Jiao Tong University

Department of

Mathematics上海交通大学数学系

Contents

Welcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Index of Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Minhang Campus Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Shanghai Metro Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Welcome

You are very much welcome to our Workshop on Algebraic Combinatorics atShanghai Jiao Tong University.

The workshop will take place from Sept. 15 to Sept. 18 and will be housed in the mathe-matics building of Shanghai Jiao Tong University. The scientific program will consist of tenlectures of 50 minutes, thirty-three lectures of 30 minutes and ten lectures of 10 minutes.All 10 minutes speakers will have the chance to give another 30 minutes presentation in oursatellite seminar on Sept. 19.

There is basically no cost and no budget. But we believe that you can find in our work-shop lots of beautiful mathematical theorems and structures. There are even no conferencebadges. But we hope that you can get to know many mathematical good friends via theirmathematics.

We will provide all participants and accompanying persons modest lunches in universitycanteen. All of you are also invited to a conference banquet on the night of Sept. 16.Please come early to register to the lunches/banquet on the conference venue and take thelunch/banquet tickets – this helps us estimate the number of seats to be reserved in thecanteen/restaurant.

Although we have a dense schedule for mathematics in the south-west corner of Shang-hai, we wish that you can still find some time to do a wonderful city tour here, especially,considering that Shanghai Tourism Festival 2011 will last from this Sept. 10 to Oct. 6.

When preparing your trip to Shanghai, it may be useful to observe the weather broadcasthere: http://news.bbc.co.uk/weather/forecast/1713

Please do not hesitate to contact us for any possible help. We thank you for coming toour workshop and we wish you a good memory here.

Workshop Organizers:

Eiichi Bannai Shanghai Jiao Tong UniversitySuogang Gao Hebei Normal UniversityJun Ma Shanghai Jiao Tong UniversityKaishun Wang Beijing Normal UniversityYaokun Wu Shanghai Jiao Tong UniversityXiao-Dong Zhang Shanghai Jiao Tong University

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Program

Sept. 15Chair:Eiichi Bannai

9:00-9:50

William J. Martin (Worcester Polytechnic Institute)Some Problems in the Theory of Q-Polynomial AssociationSchemes

Chair:Tatsuro Ito

10:00-10:50

Jianmin Ma (Hebei Normal University)Skew-Symmetric Association Schemes

Chair:Rongquan Feng

11:00-11:50

Akihiro Munemasa (Tohoku University)Frames of the Leech Lattice

11:50-13:20 Lunch

Sept. 15Chair:Yoshiaki Itoh

13:30-14:00

Mitsugu Hirasaka (Pusan National University)Construction of Algebraically Isomorphic Association Schemes

14:00-14:30

Sho Suda (Tohoku University)New Parameters of Subsets in Polynomial Association Schemes

Chair:Tongsuo Wu

14:40-15:10

Yuqun Chen (South China Normal University)Grobner-Shirshov Bases and PBW Theorems

15:10-15:40

Wei Wang (Xi’an Jiaotong University)Generalized Characteristic Polynomials and GeneralizedGM-Switchings of Graphs

Chair:Mitsugu Hirasaka

15:50-16:20

Jeong Rye Park (Pusan National University)On 3-Equivalenced Association Scheme

16:20-16:50

Nobuo Nakagawa (Kobe Gakuin University)On Non-Isomorphism Problems of Strongly Regular GraphsConstructed by p-ary Bent Functions

Chair:Jiyou Li

17:00-17:30

Yi-Huang Shen (University of Science and Technology of China)Stanley Decompositions of Monomial Ideals

17:30-18:00

Cuipo Jiang (Shanghai Jiao Tong University)On Classification of Rational Vertex Operator Algebrasof Central Charge 1

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Sept. 16Chair:Hao Shen

9:00-9:50

Keqin Feng (Tsinghua University)Cyclotomic Constructions on Codebooks, MUB’s, SIC-POVM ’sand Sphere t-Designs

9:50-10:00

Aixian Zhang (Capital Normal University)Cyclotomic Constructions of Codebooks

Chair:Jacobus Koolen

10:10-11:00

Tatsuro Ito (Kanazawa University)Finite Dimensional Irreducible Representations of Certain

Subalgebras of the Quantum Affine Algebra Uq(bsl2)Chair:Yi-Huang Shen

11:10-11:40

Hiroshi Nozaki (Tohoku University)A Characterization of Strongly Regular Graphs from EuclideanRepresentations of Graphs

11:40-13:20 Lunch

Sept. 16Chair:Beifang Chen

13:30-14:00

Hao Shen (Shanghai Jiao Tong University)Resolvable Group Divisible Designs and (k; r)-Colorings ofComplete Graphs

14:00-14:30

Hyonju Yu (Pohang University of Science and Technology)Some Construction of Regular Graphs

Chair:William J. Martin

14:40-15:10

Takuya Ikuta (Kobe Gakuin University)Nomura Algebras of Nonsymmetric Hadamard Models

15:10-15:40

Koichi Betsumiya (Hirosaki University)Even Self-dual Codes Over GF(4)

Chair:Jiayu Shao

15:50-16:20

Yoshiaki Itoh (Institute of Statistical Mathematics)Random Sequential Packing of Cubes

16:20-16:50

Shun’ichi Yokoyama (Kyushu University)Sage: Unifying Monstrous Moonshine, Modular Functions andMathematical Softwares

16:50-17:00

Houyi Yu (Shanghai Jiaotong University)Commutative Rings R Whose C(AG(R)) Only Consist ofTriangles

Chair:Yaokun Wu

17:10-17:40

Takayuki Okuda (University of Tokyo)An Analogue of Fisher Type Inequality on Compact SymmetricSpaces

17:40-17:50

Ziqing Xiang (Shanghai Jiao Tong University)Fisher Type Inequality for Boolean Designs

17:50-18:00

Eiichi Bannai (Shanghai Jiao Tong University)Some Open Problems on Various Concepts of t-Designs andFisher Type Inequalities

18:30- The Workshop Banquet

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Sept. 17Chair:Akihiro Higashitani

8:40-9:30

Beifang Chen (Hong Kong University of Science and Technology)Lattice Polytopes, Ehrhart Polynomials, and Tutte LikePolynomials Associated with Graphs

Chair:Yoshihiro Mizoguchi

9:40-10:30

Jun Ma (Shanghai Jiao Tong University)Tutte Polynomial and G-Parking Functions

Chair:Xiaodong Zhang

10:40-11:30

Rongquan Feng (Peking University)On the Coverings of Graphs

11:30-13:20 Lunch

Sept. 17Chair:Yuehui Zhang

13:30-14:00

Teruhisa Kadokami (East China Normal University)Amphicheirality of Links and Alexander Invariants

14:00-14:30

Akihiro Higashitani (Osaka University)Roots of Ehrhart Polynomials of Reflexive Polytopes Arisingfrom Graphs

Chair:Kaishun Wang

14:40-15:10

Chie Nara (Tokai University)Flat Foldings of Convex Polyhedra

15:10-15:40

Tao Feng (Zhejiang University)Recent Progress on Skew Hadarmard Difference Sets

Chair:Qiaoliang Li

15:50-16:20

Yoshihiro Mizoguchi (Kyushu University)Generalization of Compositions of Cellular Automata On Groups

16:20-16:50

Prabhu Manyem (Shanghai University)Optimization Problems and Universal Horn Formulae inExistential Second Order Logic

Chair:Tao Feng

17:00-17:30

Jun Guo (Langfang Teachers’ College)An Erdos-Ko-Rado Theorem in General Linear Groups

17:30-17:40

Yaokun Wu (Shanghai Jiao Tong University)Lit-Only σ-game On a Connected Graph with At LeastOne Loop

17:40-17:50

Yun-Ping Deng (Shanghai Jiao Tong University)Maximum-Size Independent Sets and Automorphism Groups ofTensor Powers of the Even Derangement Graphs

17:50-18:00

Kyoung-Tark Kim (Pusan National University)On Decompositions of Special Linear Algebras Over a Field ofPositive Characteristic

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Sept. 18Chair:Akihiro Munemasa

9:00-9:50

Jacobus Koolen (Pohang University of Science and Technology)On Connectivity Problems in Distance-Regular Graphs

Chair:Keqin Feng

10:00-10:50

Xavier Dahan (Kyushu University)Ramanujan Graphs of Very Large Girth Based on Octonions

Chair:Suogang Gao

11:00-11:30

Hirotake Kurihara (Tohoku University)On the Character Tables of Some Association Schemes Based ont-Singular Linear Spaces

11:30-12:00

Kaishun Wang (Beijing Normal University)A New Model for Pooling Designs

12:00-13:20 Lunch

Sept. 18Chair:Jianmin Ma

13:30-14:00

Jongyook Park (Pohang University of Science and Technology)On the Parameters of a Distance-Regular Graph

14:00-14:30

Xiao-Dong Zhang (Shanghai Jiao Tong University)The Signless Laplacian Spectral Radii of Graphs with GivenDegree Sequences

14:30-15:00

Tetsuji Taniguchi (Matsue College of Technology)On Fat Hoffman Graphs with Smallest Eigenvalue At Least −3

Chair:Jun Ma

15:10-15:40

Guofu Yu (Shanghai Jiao Tong University)Blaszak-Marciniak Lattice, Toda Lattice and CombinatorialNumbers

15:40-16:10

Stefan Grunewald (CAS-MPG Partner Institute forComputational Biology)On Agreement Forests

Chair:Stefan Grunewald

16:20-16:50

Yoshio Sano (National Institute of Informatics)The Competition Number of a Graph and the Holes inthe Graph

16:50-17:20

Jeongmi Park (Pusan National University)The Niche Graphs of Interval Orders

Chair:Tetsuji Taniguchi

17:30-17:40

Peng Li (Shanghai Jiao Tong University)A Four Sweep LBFS Algorithm for Recognizing Interval Graphs

17:40-17:50

Xiaomei Chen (Harbin Institute of Technology)On Invariants of Digraphs

17:50-18:00

Yangjing Long (Max Planck Institute for Mathematics in theSciences)Relations Between Graphs

18:00-18:30

Simone Severini (University College London)A Role for the Lovasz theta Function in Quantum Mechanics

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Abstracts

Sept. 15, 9:00 – 9:50 William J. Martin

Some Problems in the Theory of Q-Polynomial Association Schemes

William J. MartinWorcester Polytechnic Institute

Email: martin@wpi.edu

Q-polynomial, or “cometric”, association schemes were defined in 1973. Perhaps the most importantexamples are the classical distance-regular graphs. Up until 1998, very little was known about Q-polynomial schemes which are not also P -polynomial. With two fundamental papers of H. Suzuki,that changed.

The last decade has seen a flurry of activity in this area. Many new examples have been found,including the first known infinite family of primitive Q-polynomial schemes which are neither P -polynomial nor duals of P -polynomial schemes. A few surprising connections have been uncovered– connections to finite geometry and quantum information theory. And a few structure theoremshave been established, notably the “dual Bannai-Ito conjecture” where Williford and the speakerestablish that there are only finitely many Q-polynomial association schemes naturally residing inany dimension larger than two.

But many fundamental questions remain unanswered. This talk will attempt to organize theseopen questions, populating the field of important questions with perhaps more accessible preliminaryand related problems. It is hoped that the presentation will attract a few talented young researchersto this important and fertile area of mathematics.

Keywords: Association scheme; Q-polynomial; Cometric; Bose-Mesner algebra.

Sept. 15, 10:00 – 10:50 Jianmin Ma

Skew-Symmetric Association Schemes

Jianmin MaHebei Normal University

Email: jianminma@yahoo.com

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other thanthe diagonal one. Here we focus on these schemes with four association classes. It was discovered thattheir character tables fall into three types, from which their intersection matrices were determined.Then we applied these results to two particular families of association schemes: amorphous andpseudocyclic schemes. Our results show these schemes are rather restricted and interesting as well.This is a jointed work with Kaishun Wang.

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Sept. 15, 11:00 – 11:50 Akihiro Munemasa

Frames of the Leech Lattice

Akihiro MunemasaTohoku University

Email: munemasa@math.is.tohoku.ac.jp

Rowena A. L. BettyUniversity of the Philippines-Diliman

Email: ralbetty@gmail.com

In this talk, we give a precise description of Rains’ algorithm for classifying self-dual ZZ4-codeswith a given residue code. We then describe our modification of this algorithm to classify only TypeII ZZ4-codes with minimum Euclidean weight at least 16. Such a code of length 24 produces theLeech lattice, and our result gives classification of frames of the Leech lattice up to the action of theautomorphism group.

Keywords: Self-dual code; Lattice; Golay code; Extremal type II code.

Sept. 15, 13:30 – 14:00 Mitsugu Hirasaka

Construction of Algebraically Isomorphic Association Schemes

Mitsugu HirasakaPusan National University

Email: hirasaka@pusan.ac.kr

Let (X, S) be an association scheme with a closed subset T of S and Xi | i = 1, 2, . . . , m the setof cosets of T in X where m is the index of T in S. Suppose that σ1, σ2, . . . , σm ∈ Aut(T ) extendsto σ1, σ2, . . . , σm ∈ Aut(S) where σi : S → S is defined by s 7→ s if s ∈ S \ T , and s 7→ sσi ifs ∈ T . Then (X, S′) is an association scheme where S′ = s′ | s ∈ S, s′ = s if s ∈ S \ T , ands′ =

⋃mi=1 sσ

i ∩ (Xi ×Xi). In this talk we will apply this method for a class of association schemesto obtain some non-isomorphic association schemes with the same intersection numbers.

Sept. 15, 14:00 – 14:30 Sho Suda

New Parameters of Subsets in Polynomial Association Schemes

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Sho SudaTohoku University

Email: suda@ims.is.tohoku.ac.jp

In this talk, we define new parameters, a zero interval and a dual zero interval, of subsets in P - orQ-polynomial association schemes. A zero interval of a subset in a P -polynomial association schemeis a successive interval index for which the inner distribution vanishes, and a dual zero interval of asubset in a Q-polynomial association scheme is a successive interval index for which the dual innerdistribution vanishes. We derive bounds of the lengths of a zero interval and a dual zero intervalusing the degree and dual degree respectively, and show that a subset in a P -polynomial associationscheme (resp. a Q-polynomial association scheme) having a large length of a zero interval (resp. adual zero interval) induces a completely regular code (resp. a Q-polynomial association scheme).Moreover, we consider the spherical analogue of a dual zero interval.

Sept. 15, 14:40 – 15:10 Yuqun Chen

Grobner-Shirshov Bases and PBW Theorems

Yuqun ChenSouth China Normal UniversityEmail: yqchen@scnu.edu.cn

Let M be a class of algebras (variety or category with free objects) with multiple operations over afield k or a commutative algebra K over k (Ω-algebras in the sense of Higgins-Kurosh). We call theanalogy as “Composition-Diamond lemma” (CD-lemma for short), combining Newmann (Diamond)Lemma 1942 for direct graphs (rewriting systems for semigroups), Shirshov composition lemma 1962for Lie algebras and Bergman Diamond lemma for associative algebras 1978 (Bokut 1976).

M-CD-lemma Let M be a class of (in general, non-associative) Ω-algebras, M(X) a free Ω-algebra in M generated by a set X with a linear basis consisting of “normal (non-associative Ω-)words” [u], < a monomial ordering on normal words, S ⊂ M(X) a monic subset and Id(S) theideal of M(X) generated by S. Let S be a Grobner-Shirshov basis (this means that any compositionof elements in S is trivial). Then

(a) If f ∈ Id(S), then [f ] = [asb], where [f ] is the leading word of f and [asb] is a normal S-word.

(b) Irr(S) =[u]|[u] 6= [asb], s ∈ S, [asb] is a normal S − word is a linear basis of the quotientalgebra M(X|S) = M(X)/Id(S).

In many cases, each of conditions (a) and (b) is equivalent to the condition that S is a Grobner-Shirshov basis in M(X). But in some of our “new CD-lemmas”, this is not the case, for example,in the cases of conformal algebras and dialgebras.

Let the normal words N of the free algebra M(X) be a well-ordered set and 0 6= f ∈ M(X).Denote f the leading word of f . f is called monic if the coefficient of f is 1.

A well ordering on N is monomial if for any u, v, w ∈ N ,

u > v =⇒ w|u > w|v,

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where w|u = w|x7→u and w|v = w|x7→v.

Let S ⊂ M(X), s ∈ S, u ∈ N . Then, roughly speaking, u|s = u|xi 7→s, where xi is theindividuality occurrence of the letter xi ∈ X in u, is called an S-word. An S-word u|s is normal ifu|s = u|s.

Given a monic subset S ⊂M(X) and w ∈ N , an intersection (inclusion) composition h is calledtrivial modulo (S,w) if h can be presented as a linear combination of normal S-words with leadingwords less than w; a left (right) multiplication composition (or another kinds composition) h is calledtrivial modulo (S) if h can be presented as a linear combination of normal S-words with leadingwords less than or equal to h.

The set S is a Grobner-Shirshov basis in M(X) if all the possible compositions of elements inS are trivial modulo S and corresponding w.

We list some known CD-lemmas:

For Lie algebras (Shirshov 1962).

For (commutative, anti-commutative) non-associative algebras (Shirshov 1962).

For associative algebras (Shirshov 1962, Bokut 1976, Bergman 1978, Mora 1986).

For commutative algebra (Buchberger 1965).

For free Ω-algebras (Knuth, Bendix 1968).

For restricted Lie algebras (A. A. Mikhalev, 1989).

For Lie superalgebras (A. A. Mikhalev 1989).

For tensor product of a polynomial algebra and an exterior (Grassman) algebra (T. Stokes 1990,A. A. Mikhalev and A. A. Zolotykh 2007).

For associative algebras over commutative algebras (A. A. Mikhalev and A. A. Zolotykh 1998).

For associative conformal algebras (L. A. Bokut, Y. Fong and W. F. Ke 2004).

For non-commutative power series algebras (L. Hellstrom 2002).

For modules (S.-J. Kang and K.-H. Lee 2000, E. S. Chibrikov 2007).

For path algebras (D. R. Farkas, C. D. Feustel and E. L. Green 1993).

For Γ-algebras (L. A. Bokut and K. P. Shum 2007).

For algebras based on well-ordered semigroups (Y. Kobayashi).

Last years, we proved CD-lemmas for:

Lie algebras over commutative algebras, metabelian Lie algebras, pre-Lie (Vinberg) algebras, Rota-Baxter algebras, (Loday) dialgebras, associative n-conformal algebras, associative algebras withmultiple operators (associative Ω-algebras), small category algebras, tensor products of free asso-ciative algebras, associative differential algebras, associative λ-differential Ω-algebras, L-algebras,non-associative Ω-algebras over commutative algebras, S-act algebras. For details, one can refer

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to L. A. Bokut, Yuqun Chen and K. P. Shum, Some new results on Groebner-Shirshov bases,arxiv.org/abs/1102.0449.

By using CD-lemma for associative algebras, a typical application is to prove classical PBWtheorem: each Lie algebra over a field can be embedded its universal enveloping associative algebra.By using CD-lemma for di- (Rota-Baxter, pre-Lie, non-associative, resp.) algebras, we reprove thateach Leibniz (dendriform, Lie, Akivis, resp.) algebra can be embedded its universal enveloping di-(Rota-Baxter, pre-Lie, non-associative, resp.) algebra.

Recently, we establish a CD-lemma for Lie algebras over commutative algebras. We prove Cohn’sconjecture is valid when p = 2, 3, 5 and we present an algorithm that one can check for any p, whetherCohn’s conjecture is valid or not.

Cohn’s conjecture (1963) Let K = k[y1, y2, y3|ypi = 0, i = 1, 2, 3] be an algebra of truncated

polynomials over a field k of characteristic p > 0. Let

Lp = LieK(x1, x2, x3 | y3x3 = y2x2 + y1x1).

Then Lp can not be embedded its universal enveloping associative algebra.

Sept. 15, 15:10 – 15:40 Wei Wang

Generalized Characteristic Polynomials and Generalized GM-Switchings of Graphs

Wei WangXi’an Jiaotong University

Email: wang_weiw@163.com

Despite the fact that there exist pairs of cospectral and non-isomorphic graphs, the spectra ofgraphs plays an important role in spectral graph theory. There are abundant work concerning theconstruction of cospectral graphs in the literature. For example, one powerful tool to generatecospectral graphs was introduced by Godsil and McKay in 1982, which is now known as the GM-switching method.

In this talk, we are mainly concerned with the the problem “To what extent does the converse tothe GM-switching still hold?” We shall consider a generalization of the GM-switching method andinvestigate the relationship between the generalized GM-switchings and the generalized characteristicpolynomials of graphs with some partial answers provided.

Sept. 15, 15:50 – 16:20 Jeong Rye Park

On 3-Equivalenced Association Scheme

Mitsugu Hirasaka

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Pusan National UniversityEmail: hirasaka@pusan.ac.kr

Kyoung-tark KimPusan National University

Email: poisonn00@pusan.ac.kr

Jeong Rye ParkPusan National University

Email: parkjr@pusan.ac.kr

Let (Ω, S) be an association scheme where Ω is a finite set and S is a partition of Ω × Ω. For apositive integer k we say that (Ω, S) is k-equivalenced if each nonidentity element of S has valencyk. In this talk we consider a 3-equivalenced association scheme (Ω, S) and show that S is the setof orbits of a Frobenius group. We also mention basic results of 4-equivalened association schemes.Every 4-equivalened association scheme is a symmetric scheme (It was proved by Z. Arad, Y. Erezand M. Muzychuk) and we prove that for any s ∈ S, the complex product ss contains at most threeelements.

Sept. 15, 16:20 – 16:50 Nobuo Nakagawa

On Non-isomorphism Problems of Stronly Regular Graphs Constructed by p-aryBent Functions

Nobuo NakagawaUniversity of Kinki

Email: nakagawa@math.kindai.ac.jp

We construct a graph Γ(f, p) by using a p-ary bent function f from GF (p2k) to GF (p) as thefollowing. Let S be the non-zero squares of GF (p). The vertices set is GF (p2k) and a vertex x isadjacent to a vertex y if and only if f(x− y) ∈ S. Then it is proved that Γ(f, p) is strongly regulargraphs under some condition (A) by Chee, Tan and Zhang. In below we suppose k = 2 and setF := GF (p4). Then Γ(f, p) is SRG with parameters (p(p2+1)(p−1)

2 , p(p2+3)(p−2)4 , p(p2−p+2)(p−1)

4 ) if fsatisfies (A) from the above.

Now there are two interesting bent functions satisfying (A), one of them is f0(x) := Tr(x2) andanother one is g0(x) := Tr(x2 + xp3+p2−p+1) which is constructed by Helleseth and Kholosha.

Theorem 1 Let p be an odd prime which is less than 20. Then the graph Γ(f0, p) is not isomorphicto the graph Γ(g0, p).

The outline of the proof is the following. The automorphism groups of Γ(f0, p) and Γ(g0, p) containthe translation group T := ta|a ∈ F and the scalar multiplication group M := mα|α ∈ GF (p)×where ta(x) = x + a and mα(x) = αx. Besides Aut(Γ(f0, p)) contains the orthogonal group G :=O−(F ) of minus type with respect to a bilinear mapping b(x, y) := Tr(xy). We note f0(x) = b(x, x).Then F has a orthgonal basis ui|1 ≤ i ≤ 4 such that f0(ui) = 1 for i = 1, 2, 3 and f0(u4) = γ0 fora fixed γ0 6∈ S.

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Set Ω(α, β) := y =∑4

i=1 βiui ∈ F |f0(y) = α, β = β1 for α ∈ S and β ∈ F. It hold that

T is transitive on F · · · (1);

< G, M > is transitive on the 1-st neighborhood of 0 · · · (2);

Gu1 (the stabilizer of u1) is transitive on Ω(α, β) for each α ∈ S, β ∈ F · · · (3).

For a triangle ∆(a, b, c), we denote the cardinality of v ∈ F |v is adjacent to a, b and c byN(∆(a, b, c)). We take a triangle ∆(a′, b′, c′) of Γ(g0, p). If there is an isomorphism ψ from Γ(g0, p)to Γ(f0, p), then we may asumme ψ(a′) = 0 from (1), ψ(b′) = u1 from (2) and ψ(c′) = yα,β for yα,β ∈Ω(α, β) for some α and β from (3). Therefore if N(∆(a′, b′, c′)) 6∈ N(∆(0, u1, yα,β))|α ∈ S, β ∈ Ffor a certain ∆(a′, b′, c′) of Γ(g0, p), it means the non-isomorphism between Γ(f0, p) and Γ(g0, p). Weused Magma to compute N(∆(0, u1, yα,β)) for each α ∈ S, β ∈ F.

Keywords: Strongly regular graph; P -ary bent function; Orthogonal group; Partial difference set.

Sept. 15, 17:00 – 17:30 Yi-Huang Shen

Stanley Decompositions of Monomial Ideals

Yi-Huang ShenUniversity of Science and Technology of China

Email: yhshen@ustc.edu.cn

In 1982, Richard Stanley conjectured that, for a finitely generated Zn-graded module M , its Stanleydepth is bounded below by its depth: sdepth(M) ≥ depth(M). This conjecture is a stronger versionof his another well-known conjecture: Cohen-Macaulay simplicial complexes are partitionable. TheStanley conjecture has been confirmed in several special cases, but still remains widely open. Oneobstacle of verifying this conjecture lies in the difficulty of computing the Stanley depth. In thistalk, we plan to go over some recent progress around this Stanley conjecture. Especially, we willfocus on those of the Stanley decompositions of monomial ideals.

Keywords: Stanley conjecture; Stanley decomposition; Monomial ideals.

Sept. 15, 17:30 – 18:00 Cuipo Jiang

On Classification of Rational Vertex Operator Algebras of Central Charge 1

Cuipo JiangShanghai Jiao Tong UniversityEmail: cpjiang@sjtu.edu.cn

This talk is devoted to recent progress in classification of rational vertex operator algebras of centralcharge 1, including a new result we have obtained.

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Sept. 16, 9:00 – 9:50 Keqin Feng

Cyclotomic Constructions on Codebooks, MUB’s, SIC-POVM ’s and Spheret-Designs

Keqin FengTsinghua University

Email: kfeng@math.tsinghua.edu.cn

Aixian ZhangCapital Normal University

Email: zhangaixian1008@126.com

Cyclotomy over finite fields and related number-theory machineries, Gauss and Jacobi sums, are oneof the important methods to construct a variety of combinatorial designs. In this talk we considerhow far the cyclotomic method can go to construct codebooks used in communication systems,MUB’s and SIC-POVM’s used in quantum computation and complex sphere 2-designs reaching orapproximately reaching the Welch bound.

Sept. 16, 9:50 – 10:00 Aixian Zhang

Cyclotomic Constructions of Codebooks

Keqin FengTsinghua University

Email: kfeng@math.tsinghua.edu.cn

Aixian ZhangCapital Normal University

Email: zhangaixian1008@126.com

An (N, K) codebook C is a set c1, · · · , cN of N unit norm complex vectors ci (1 ≤ i ≤ N) in CK .Codebooks meeting the Welch bound are used in the direct spread CDMA systems to distinguishamong the signals of different users. Many series of codebooks were constructed by using thedifference sets and almost difference sets meeting or nearly meeting the Welch bound.

In this report, we will show more series of codebooks which were constructed by using Gausssums, Jacobi sum and cyclotomic numbers as main tools and this construction under looser conditionthan ones required by DS and ADS.

Keywords: Codebook; Gauss sum; Cyclotomic number; Almost difference set.

Sept. 16, 10:10 – 11:00 Tatsuro Ito

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Finite Dimensional Irreducible Representations of Certain Subalgebras of the

Quantum Affine Algebra Uq(sl2)

Tatsuro ItoKanazawa University

Email: tatsuro@kenroku.kanazawa-u.ac.jp

As an application of the classification of generic TD-pairs, we determine finite dimensional irreduciblerepresentations of certain subalgebras of the quantum affine algebra Uq(sl2). In doing so, we revisitthe Terwilliger algebras of P- and Q-polynomial schemes and reprove the classical result of Chari-Pressley that determine finite dimensional irreducible representations of Uq(sl2). This is a joint workwith my student, Tomoya Yattai.

Keywords: Quantum affine algebra; TD-pair; P- and Q-polynomial scheme.

Sept. 16, 11:10 – 11:40 Hiroshi Nozaki

A Characterization of Strongly Regular Graphs from Euclidean Representations ofGraphs

Hiroshi NozakiTohoku University

Email: nozaki@ims.is.tohoku.ac.jp

Let G be a simple graph with n vertices. G is representable in Rd if there is an embedding of thevertex set in Rd and distinct non-negative constants a and b such that for all vertices u and v,

||u− v|| =

a, if u ∼ v;b, otherwise,

where ||x|| =√

xT x. This embedding in Rd is called an Euclidean representation of G. Anyrepresentation of G is a two-distance set, that is, the set has only two distances a and b betweendistinct points. By scaling we can normalize a = 1. Einhorn–Schoenberg (1966) proved that there isan Euclidean representation whose dimension is smaller than n−1. Recently Roy (2010) determinesthe embedding dimension by some values about the eigenspaces of the adjacency matrix of G. Inthis talk, we introduce the embedding theory due to Einhorn–Schoenberg and Roy, and the basictheory on s-distance sets. As a new result we clarify when the embedding is on a sphere. Moreoverwe give a new characterization of strongly regular graphs from the spherical representations of G.This is a joint work with Masashi Shinohara.

Sept. 16, 13:30 – 14:00 Hao Shen

Resolvable Group Divisible Designs and (k, r)-Colorings of Complete Graphs

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Hao ShenShanghai Jiao Tong UniversityEmail: haoshen@sjtu.edu.cn

A (k, r)-coloring of a complete graph K is a coloring of the edges of K with r colors such that allmonochromatic connected subgraphs have at most k vertices. The Ramsey number f(k, r) is definedto be the smallest u such that the complete graph K with u vertices does not admit a (k, r)-coloring.In this talk, we study applications of resolvable group divisible designs in determining the Ramseynumber f(k, r) and obtain some new progresses.

Sept. 16, 14:00 – 14:30 Hyonju Yu

Some Construction of Regular Graphs

Hyonju YuPohang University of Science and Technology

Email: lojs4110@postech.ac.kr

For a given graph G and a positive integer r, we construct a (dmax(G)+r−1)-regular graph G(r) suchthat λmin(G(r)) ∈ [λmin(G) − 1, λmin(G)] and limr→∞ λmin(G(r)) = λmin(G) − 1 where λmin(G)is the smallest eigenvalue of G. As applications, we obtain the following theorems.

Theorem 2 Let γregk be the supremum of the smallest eigenvalues of k-regular graphs with smallest

eigenvalue < −2. Then limk→∞ γregk = −1−√2.

Theorem 3 Let δregk be the supremum of the smallest eigenvalues of k-regular graphs with smallest

eigenvalue < −1 − √2. Then limk→∞ γreg

k = δ where δ is the smallest root(≈ −2.4812) of thepolynomial x3 + 2x2 − 2x− 2.

Theorem 4 Every number in the interval (−∞,−√

2 +√

5 − 1] is a limit point of the smallesteigenvalues of regular graphs.

Keywords: Regular graph; Smallest eigenvalue; Limit point graph.

Sept. 16, 14:40 – 15:10 Takuya Ikuta

Nomura Algebras of Nonsymmetric Hadamard Models

Takuya IkutaKobe Gakuin University

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Email: ikuta@law.kobegakuin.ac.jp

Akihiro MunemasaTohoku University

Email: munemasa@math.is.tohoku.ac.jp

Jaeger and Nomura constructed nonsymmetric Hadamard models for link invariants from Hadamardmatrices. These models are closely related to the Hadamard model originally constructed by Nomura.

In this talk, we explicitly construct nonsymmetric association schemes derived from the Hadamardgraphs. Also, we show that the Bose-Mesner algebras of these association schemes coincide with theNomura algebras of the nonsymmetric Hadamard models.

Keywords: Association schemes; Hadamard matrix; Spin models.

Sept. 16, 15:10 – 15:40 Koichi Betsumiya

Even Self-dual Codes Over GF(4)

Koichi BetsumiyaHirosaki University

Email: betsumi@cc.hirosaki-u.ac.jp

It is known that any even code of rate 1/2 over GF(4) has to be an Hermitian self-dual code byGleason-Pierce-Ward theorem. In this talk, we give some properties of the class of Euclidean self-dual even codes in the even codes of rate 1/2 by using the invariant theory of complete weightenumerators.

Keywords: Gleason-Pierce-Ward theorem; Euclidean self-dual codes; Hermitian self-dual codes.

Sept. 16, 15:50 – 16:20 Yoshiaki Itoh

Random Sequential Packing of Cubes

Mathieu Dutour SikiricInstitut Rudjer Boscovic, Zagreb

Email: mdsikir@irb.hr

Yoshiaki ItohInstitute of Statistical Mathematics, Tachikawa, Tokyo

Email: itoh@ism.ac.jp

Consider the random sequential packing of cubes of edge length 1 in a parallel position in a largercube of edge length x. For the d-dimensional case the limit packing density is conjectured to be

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equal to βd as x → ∞, where β is the limit packing density for d = 1 given by Renyi (1958). Thecomputer simulations do not support the conjecture. Consider the simplest model. Cubes of edgelength 2 are put sequentially at random into the cube of edge length 4, with a cubic grid with length1. The packing density γd at the saturation of dimension d obtained by computer simulations fitsto the power law (Itoh and Ueda (1983)). It is shown to be larger than (3

2 )d (Poyarkov (2003))mathematically. Consider the simplest random sequential packing into torus. The case d = 1, 2gives the tiling of cubes, while the case 3 ≤ d does not always give the tiling (Dutour Sikiric, Itohand Poyarkov (2007)). The result is extended to continuous version of the problem (Dutour Sikiricand Itoh (2010)).

References

[1] Dutour Sikiric, M. and Itoh, Y. (2011) Random Sequential Packing of Cubes, World Scientific.

Sept. 16, 16:20 – 16:50 Shun’ichi Yokoyama

Sage: Unifying Monstrous Moonshine, Modular Functions and MathematicalSoftwares

Shun’ichi YokoyamaKyushu University

Email: s-yokoyama@math.kyushu-u.ac.jp

Sage is a free open-source mathematics software system licensed under the GPL, and its mission-statement is creating a viable free open source alternative to Magma, Maple, Mathematica andMatlab. It enables us to compute various mathematical objects. In this talk, we first explain how touse Sage and what Sage can do. After that, we give a demonstration of Sage and some applicationsfor the research of algebraic combinatorics that is strongly related to number theory and algebra. Inparticular, we show examples of computation to observe ”Monstrous Moonshine” and related topicsnumerically.

Sept. 16, 16:50 – 17:00 Houyi Yu

Commutative Rings R Whose C(AG(R)) Only Consist of Triangles

Tongsuo Wu, Houyi YuShanghai Jiaotong University

Email: tswu@sjtu.edu.cn,yhy178@163.com

Let R be a commutative ring. The set I(R) of all ideals of R is a po-semiring under the multipli-cation, the addition and the usual inclusion of ideals. The zero-divisor graph of I(R) is called theannihilating-ideal graph of R, denoted by AG(R). We write G for the set of graphs whose coresonly consist of triangles. We first determine the type of the graphs in G which can be realizedas the zero-divisor graph of po-semirings and the annihilating-ideal graph of commutative rings,respectively. Then we give a necessary and sufficient condition for R with AG(R) ∈ G. Finally, acomplete characterization in terms of the quotient of polynomial rings is established for a finite ringR with AG(R) ∈ G. Also, the connections between the finite rings and the corresponding graphs isrealized.

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Keywords: Commutative ring; Po-semirings; Zero-divisors; Annihilating-ideals; Triangles; Fan graphs;Horns.

Sept. 16, 17:10 – 17:40 Takayuki Okuda

An Analogue of Fisher Type Inequality on Compact Symmetric Spaces

Takayuki OkudaUniversity of Tokyo

Email: okuda@u-tokyo.ac.jp

Fisher type inequality for spherical designs is important in the meaning of giving a lower bound of t-designs, an upper bound of s-distance sets and relation between designs and codes on sphere. We areinterested in finding generalization of it on compact symmetric spaces. In the case where the compactsymmetric space has rank one, a similar theory was showed by Bannai–Hoggar (1985). In the case forhigher rank, Bachoc–Bannai–Coulangeon (2004) showed an analogue on real Grassman manifolds.Furthermore, similar results on complex Grassman manifolds (Roy 2010) and on unitary groups (Roy2009) were known. Roy and Suda (2011) also showed a similar result on complex spheres(which isnot symmetric spaces). In this talk, we define designs and codes on general compact symmetricspaces by using the representation theory, and show an analogue of Fisher type inequality.

Keywords: Compact symmetric space; Spherical design; Spherical code; Fisher type inequality.

Sept. 16, 17:40 – 17:50 Ziqing Xiang

Fisher Type Inequality for Boolean Designs

Ziqing XiangShanghai Jiao Tong University

Email: ziqingxiang@gmail.com

In this talk, we consider a block matrix whose blocks are non-zero set-inclusion matrices. Wecalculate its rank over real field and then, as an application, establish a Fisher type inequality forBoolean designs, thus answering an open problem of Delsarte and Seidel posed in 1989.

Keywords: Fisher type inequality; Boolean design; Incidence matrix; Rank; Totally positive.

Sept. 16, 17:50 – 18:00 Eiichi Bannai

Some Open Problems on Various Concepts of t-Designs and Fisher Type Inequalities

Eiichi BannaiShanghai Jiaotong UniversityEmail: bannai@sjtu.edu.cn

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We discuss the similarity between ”spherical t-designs and Euclidean t-designs” and ”combinatorialt-(v, k, λ) designs and regular t-wise balanced designs (i.e., combinatorial t-designs with differentblock sizes being allowed)”. Ziqing Xiang’s recent work gives a Fisher type inequality for regular t-wise balanced designs. In this talk, I want to discuss some of my so far mostly unsuccessful attemptsto try to get similar results under more general circumstances.

Sept. 17, 9:00 – 9:50 Beifang Chen

Lattice Polytopes, Ehrhart Polynomials, and Tutte Like Polynomials Associatedwith Graphs

Beifang ChenHong Kong University of Science and Technology

Email: mabfchen@ust.hk

One of the most interesting and important polynomials associated with graphs is the Tutte polyno-mial, which is a common generalization of the chromatic polynomial and the flow polynomial. It iswell-known that the values of the chromatic polynomial and the flow polynomial have combinatorialinterpretations at zero and negative integers. However, for a long time there is no such combinatorialinterpretation for the Tutte polynomial in general until recently the work of Chang, Ma, and Yehby G-parking functions and of myself by geometric method. In this talk I shall present a geometricapproach that gives rise automatically to the combinatorial interpretations for the Tutte like poly-nomials associated with graphs, including the Tutte polynomial. The approach is related to latticepolytopes, Ehrhart polynomials, and integral hyperplane arrangements.

Sept. 17, 10:00 – 10:50 Jun Ma

Tutte Polynomial and G-Parking Functions

Hungyung ChangNational Sun Yat-sen University

Email: changhy@math.nsysu.edu.tw

Jun MaShanghai Jiao Tong University

Email: majun904@sjtu.edu.cnm

Yeong-Nan YehInstitute of Mathematics, Academia Sinica

Email: mayeh@math.sinica.edu.tw

For every undirected graph, Tutte [W. T. Tutte, A contribution to the Theory of Chromatic Poly-nomials, Canad. J. Math. 6 (1953) 80-91] defined a polynomial TG(x; y) in two variables whichplays an important role in graph theory. This polynomial is called Tutte polynomial. It containsinformation about how the graph is connected. For example, TG(1; 1) is the number of spanningtrees in G, TG(2; 1) is the number of spanning forests in G. As universality of graph language,Tutte polynomial contains several famous other specialisations from other sciences such as the Jonespolynomial from knot theory and the partition functions of the Potts model from statistical physics.

Many combinatorial interpretations for TG(x, y) were made from various viewpoints. Parkingfunctions are a famous combinatorial models and have very close relations with Tutte polynomial.Konheim and Weiss [A. G. Konheim, B. Weiss, An Ocuupancy Discipline and Applications, SIAM

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J. Appl. Math. 14 (1966) 1266-1274] introduced the concept of parking functions in the study ofthe linear probes of random hashing function. Postnikov and Shapiro [A. Postnikov, B. Shapiro,Trees, Parking Functions, Syzygies, and Deformatioins of Monomial Ideals, Trans. Amer. Math.Soc. 356 (2004) 3109-3142] introduced G-parking functions in the study of certain quotients of thepolynomial ring.

In this talk, we introduce a new combinatorial interpretation for the Tutte polynomial fromviewpoint of G-parking functions.

Keywords: Parking function; Tutte Polynomial.

Sept. 17, 11:00 – 11:50 Rongquan Feng

On the Coverings of Graphs

Rongquan FengPeking University

Email: fengrq@math.pku.edu.cn

In this talk, the (regular) covering of a graph is introduced. The combinatorial construction ofcoverings of a graph is given. The isomorphism classes of some kind of coverings are enumerated.The characteristic polynomials of coverings of a graph are computed. Some problems related to thecoverings will be proposed.

Keywords: Graph covering; Voltage assignment; Isomorphism of coverings; Graph bundle.

Sept. 17, 13:30 – 14:00 Teruhisa Kadokami

Amphicheirality of Links and Alexander Invariants

T. KadokamiEast China Normal University

Email: mshj@math.ecnu.edu.cn

A. KawauchiOsaka City University

Email: kawauchi@sci.osaka-cu.ac.jp

Amphicheirality (= non-chirality) is a special symmetry for a link (in the 3-sphere). We provedsome vanishing properties for the Alexander polynomials of amphicheiral links, and applied them tomake the complete table of amphicheiral links with the crossing number up to 9. For the proof, weused invariants deduced from the t-symmetric pairing of the Alexander module.

Sept. 17, 14:00 – 14:30 Akihiro Higashitani

Roots of Ehrhart Polynomials of Reflexive Polytopes Arising from Graphs

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Akihiro HigashitaniOsaka University

Email: a-higashitani@cr.math.sci.osaka-u.ac.jp

Let P ⊂ RN be an integral convex polytope of dimension D. Given a positive integer n, we definethe numerical function i(P, n) by setting

i(P, n) = |nP ∩ ZN |.The systematic study of i(P, n) originated by Ehrhart, who established the fundamental property,which is that i(P, n) is a polynomial in n of degree D with i(P, 0) = 1. (Thus, in particular, i(P, n)can be defined for every integer n, more generally, for every complex number n.) We say that i(P, n)is the Ehrhart polynomial of P.

For an integral convex polytope P ⊂ RD of dimension D, we say that P is a reflexive polytope ifthe origin of RD is a unique integer point belonging to the interior of P and the dual polytope P∨of P is also integral, where P∨ = x ∈ RD : 〈x, y〉 ≤ 1, y ∈ P.

Let P ⊂ RD be an integral convex polytope and i(P, n) its Ehrhart polynomial. Then P isunimodularly equivalent to a reflexive polytope if and only if i(P, n) has a special property:

i(P, n) = (−1)Di(P,−n− 1).

From this functional equation, the roots of the Ehrhart polynomials of reflexive polytopes distributesymmetrically in the complex plane with respect to the vertical line <(z) = −1/2. Thus, in partic-ular, if D is odd, then −1/2 is a root.

In this talk, we introduce reflexive polytopes arising from finite graphs, which are called sym-metric edge polytopes, and we discuss the roots of their Ehrhart polynomials by using the languagesof finite graphs. Especially, we consider the following fascinating problem.

Question: What kind of graphs constructs the reflexive polytope whose Ehrhart polynomialsatisfies that all its roots have the real parts −1/2 ?

Keywords: Ehrhart polynomial; δ-vector; Reflexive polytope; Finite graph.

Sept. 17, 14:40 – 15:10 Chie Nara

Flat Foldings of Convex Polyhedra

Jin-ichi ItohKumamoto University

Email: j-itoh@kumamoto-u.ac.jp

Chie NaraTokai University

Email: cnara@ktmail.tokai-u.jp

Costin VılcuInstitute of Mathematics of the Romanian Academy

Email: Costin.Vilcu@imar.ro

Flat folding of a polyhedron is a folding by creases into a multilayered planar shape. Our problem isto flatten a convex polyhedron continuously whithout tearing nor stretching the surface. We showthat each Platonic polyhedron can be continuously flat folded onto an original face, which is provedby proposing a key lemma on rhombi.

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For general convex polyhedra, we show that every convex polyhedron possesses infinitely manycontinuous flat folding processes, which is proved by using cut loci. Moreover, we give a sufficientcondition under which every flat folded state of a convex polyhedron can be reached by a continuousfolding process. This is a partial answer to the open problem by E. Demaine et al. that every flatfolded state of a polyhedron can be reached by a continuous folding process.

Keywords: Flat folding; Convex polyhedron; Crease pattern.

Sept. 17, 15:10 – 15:40 Tao Feng

Recent Progress on Skew Hadarmard Difference Sets

Tao FengZhejiang University

Email: pku.tfeng@yahoo.com.cn

A difference set D in a finite multiplicative group G is called skew Hadamard if G is the disjointunion of D, D−1, and 1, where D−1 = d−1 : d ∈ D. The primary example (and for many years,the only known example in abelian groups) of skew Hadamard difference sets is the classical Paleydifference set in (Fq; +) consisting of the nonzero squares of Fq, where Fq is the finite field of orderq, and q is a prime power congruent to 3 modulo 4. Skew Hadamard difference sets are currentlyunder intensive study. I will give a brief survey of recent results on this topic, and report our recentconstruction using unions of cyclotomic classes. This is joint work with Qing Xiang.

Sept. 17, 15:50 – 16:20 Yoshihiro Mizoguchi

Generalization of Compositions of Cellular Automata on Groups

Yoshihiro MizoguchiKyushu University

Email: ym@imi.kyushu-u.ac.jp

Mitsuhiko FujioKyushu Institute of Technology

Email: fujio@ces.kyutech.ac.jp

Shuichi InokuchiKyushu University

Email: inokuchi@math.kyushu-u.ac.jp

A kind of notions of compositions was investigated by Sato (1994) and Manzini (1998) for linear cel-lular automata, we extend the notion to general cellular automata on groups and investigated theirproperties. We also introduce the notion of ‘Union’ and ‘Division’ of cellular automata on groups.We enumerate the all unions and compositions generated by one-dimensional 2-neighborhood cellu-lar automata over Z2 including non-linear cellular automata. Next we prove that the compositionis right-distributive over union, but is not left-distributive. Finally, we conclude by showing refor-mulation of our definition of cellular automata on group which admit more than three states. Wealso show our formulation contains the representation using formal power series for linear cellularautomata in Manzini (1998).

Keywords: Cellular automata; Groups; Models of computation; Automata.

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Sept. 17, 16:20 – 16:50 Prabhu Manyem

Optimization Problems and Universal Horn Formulae in Existential Second OrderLogic

Nerio BorgesUniversidad Simon BolıvarEmail: nborges@usb.ve

Prabhu ManyemShanghai University

Email: prabhu.manyem@gmail.com

We consider decision problems derived from optimization problems. For problems in this category,we show that they cannot be expressed in existential second order (ESO) logic with a first orderpart that is universal Horn (or simply, ESO Π1 Horn) at the structure level, as opposed to machinelevel. This is true even when we consider ordered structures. The machine level expressions ex-press the computation of a property, whereas the structure level expressions deal directly with themathematical structure. We show that an NP-complete problem (Maximum Independent Set) anda polynomially solvable problem (Maximum Matching) have identical ESO Π1 Horn expressions.

Sept. 17, 17:00 – 17:30 Jun Guo

An Erdos-Ko-Rado Theorem in General Linear Groups

Jun GuoLangfang Teachers’ CollegeEmail: guojun_lf@163.com

Kaishun WangBeijing Normal University

Email: wangks@bnu.edu.cn

Let Sn be the symmetric group on n points. Deza and Frankl [M. Deza and P. Frankl, On themaximum number of permutations with given maximal or minimal distance, J. Combin. TheorySer. A 22 (1977) 352–360] proved that if F is an intersecting set in Sn then |F| ≤ (n − 1)!. Inthis note we consider the q-analogue version of this result. Let Fn

q be the n-dimensional row vectorspace over a finite field Fq and GLn(Fq) the general linear group of degree n. A set Fq ⊆ GLn(Fq)is intersecting if for any T, S ∈ Fq there exists a non-zero vector α ∈ Fn

q such that αT = αS. LetFq be an intersecting set in GLn(Fq). We show that |Fq| ≤ q(n−1)n/2

∏n−1i=1 (qi − 1).

Keywords: Erdos-Ko-Rado theorem; General linear group.

Sept. 17, 17:30 – 17:40 Yaokun Wu

Lit-Only σ-game On a Connected Graph with At Least One Loop

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Yaokun Wu, Ziqing XiangShanghai Jiao Tong University

Email: ykwu@sjtu.edu.cn(YW),ziqingxiang@gmail.com(ZX)

Consider a graph each of whose vertices is either in the ON state or in the OFF state and call theresulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph. Aregular move at a vertex changes the states of the neighbors of that vertex and hence sends thecurrent configuration to another one. A valid move is a regular move at an ON vertex.

The σ-game on a graph G is the discrete dynamical system whose phase space PS(G) consistsof all configurations on G and the phase transitions are given by all regular moves; The lit-onlyσ-game on a graph G is the discrete dynamical system whose phase space PS∗(G) consists of allconfigurations on G and the phase transitions are given by all valid moves. To understand thereachability property of PS(G) is simply to test whether or not a binary vector lies in a given binarysubspace. The reachability property of PS∗(G) looks to be quite nonlinear and it seems much moredifficult to understand.

When the graph G is connected and has at least one loop, Goldwasser, Wang and Wu proposea conjecture on a surprisingly clean connection between PS(G) and PS∗(G) [1, Conjecture 6]. Aftera long chain of proofs as well as some verification by computer programming, we can now explicitlydescribe the reachability property of PS∗(G) in terms of that of PS(G). In particular, this workimplies that the conjecture of Goldwasser, Wang and Wu is true. Our work relies on a forbiddensubgraph characterizations of line graphs and the statement of our main result also involves linegraphs, which may be a bit strange at first sight.

References

[1] J. Goldwasser, X. Wang, Y. Wu, Minimum light numbers in the σ-game and lit-only σ-game on unicyclic and grid graphs, Electronic Journal of Combinatorics, (2011), to appear.

Sept. 17, 17:40 – 17:50 Yun-Ping Deng

Maximum-Size Independent Sets and Automorphism Groups of Tensor Powers ofthe Even Derangement Graphs

Yun-Ping Deng, Xiao-Dong ZhangShanghai Jiaotong University

Email: dyp612@hotmail.com(YD),xiaodong@sjtu.edu.cn(XZ)

Let An be the alternating group of even permutations of X := 1, 2, · · · , n and En the set of evenderangements on X. Denote by AΓq

n the tensor product of q copies of AΓn, where the Cayley graphAΓn := Γ(An, En) is called the even derangement graph. In this paper, we intensively investigatethe properties of AΓq

n including connectedness, diameter, independence number, clique number,chromatic number and the maximum-size independent sets of AΓq

n. By using the result on themaximum-size independent sets of AΓq

n, we completely determine the full automorphism groups ofAΓq

n.

Keywords: Automorphism group; Cayley graph; Tensor product; Maximum-size independent sets;Alternating group.

Sept. 17, 17:50 – 18:00 Kyoung-Tark Kim

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On Decompositions of Special Linear Algebras over a Field of Positive Characteristic

Kyoung-Tark KimPusan National University

Email: poisonn00@hanmail.net

It is well-known that for each prime power n the special linear algebra sl(n,C) has an OD (orthogonaldecomposition) into CSAs (Cartan subalgebras) with respect to the Killing form. But, it is still openwhether sl(n,C) has such an OD unless n is a prime power. So, it is natural to ask what happens ifwe consider sl(n, F ) with charF > 0. In this modular case, sl(n, F ) is a restricted Lie algebra whosename and concept are first introduced by N. Jacobson. In a restricted Lie algebra L, a torus J is anabelian p-subalgebra of L such that JK has no nonzero p-nilpotent elements where K is an algebraicclosure of F and JK := K ⊗F J . There is a fact that a CSA is the centralizer of a maximal torus ina restricted Lie algebra, but we will show that a maximal torus is itself a CSA in sl(n, F ). We aimto know whether sl(n, F ) has a direct sum decomposition into maximal tori, and when componentsin the decomposition are pairwise orthogonal with respect to the trace form.

Sept. 18, 9:00 – 9:50 Jacobus Koolen

On Connectivity Problems in Distance-Regular Graphs

Jacobus KoolenPohang University of Science and Technology

Email: koolen@postech.ac.kr

In this talk I will give new and old results on connectivity problems in distance-regular graphs.

Sept. 18, 10:00 – 10:50 Xavier Dahan

Ramanujan Graphs of Very Large Girth Based on Octonions

X. DahanKyushu University, Japan

Email: dahan@math.kyushu-u.ac.jp

J.-P. Tillich“Secret” project of INRIA-Rocquencourt, France

Email: jean-pierre.tillich@inria.fr

A k-regular connected graph is Ramanujan graph if the second largest eigenvalue in absolute value |λ|of its adjacency matrix is very small, namely verifies |λ| ≤ k−2

√k − 1 (this is indeed “essentially” the

smallest possible). They were famously introcuded by Lubotzky-Philips-Sarnak and independentlyby Margulis in the mid 80’s, and received a lot of attention. Basically, this is due to their relationshipwith expander graphs, which have many applications in Computer Science and nowadays in PureMathematics as well.

We present another construction of infinite families of Ramanujan graphs. They hold a combi-natorial property that improves upon what could be reached by any other previous constructions:the girth is very large, and actually surprisingly close to a (trivial) upper bound that was thought to

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be loose. During the talk, we will discuss the algebraic-combinatorial aspects of this construction,which used the arithmetic of octonions. We will also mention some remaining open problems.

Keywords: Ramanujan graph; Expander graph; High girth; Octonion; Moufang loop.

Sept. 18, 11:00 – 11:30 Hirotake Kurihara

On the Character Tables of Some Association Schemes Based on t-Singular LinearSpaces

Hirotake KuriharaTohoku University

Email: sa9d05@math.tohoku.ac.jp

In this talk, I talk about association schemes based on t-singular linear spaces and its charactertables for some cases. These association schemes are generalizations of association schemes basedon attenuated spaces.

Sept. 18, 11:30 – 12:00 Kaishun Wang

A New Model for Pooling Designs

Jun GuoLangfang Teacher’s CollegeEmail: guojun_lf@163.com

Kaishun WangBeijing Normal University

Email: wangks@bnu.edu.cn

In this talk we shall introduce a new model for pooling designs.

Keywords: Pooling design; Error-correcting; Johnson graph.

Sept. 18, 13:30 – 14:00 Jongyook Park

On the Parameters of a Distance-Regular Graph

Jongyook ParkPohang University of Science and Technology

Email: jongyook@postech.ac.kr

In this talk, I will discuss two results.

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1: For fixed c ≥ 1, there are finitely many distance-regular graphs with k2k ≤ c and D ≥ 6. (This

is a joint work with Jack Koolen and Greg Markowsky.)

2: Let Γ be a distance-regular graph with valency k ≥ 3 and diameter D ≥ 4. If Γ has aquadrangle, then c2 ≤ 2

D k with equality if and only if D ≥ 5 and Γ is a k-cube or D = 4 and Γ is aHadamard graph. (This is a joint work with Jack Koolen.)

Also, I will discuss extensions of these two results.

Keywords: Distance-regular graphs; Taylor graphs; Hypercubes; Hadamard graphs; Inequalities onintersection numbers.

Sept. 18, 14:00 – 14:30 Xiao-Dong Zhang

The Signless Laplacian Spectral Radii of Graphs with Given Degree Sequences

Xiao-Dong ZhangShanghai Jiao Tong UniversityEmail: xiaodong@sjtu.edu.cn

In this talk, we consider the following question ”for a given graphic degree sequence π, let

Gπ = G | G is connected with π as its degree sequence.Find the upper (lower) bounds for the Laplacian spectral radius of all graphs G in Gπ and characterizeall extremal graphs which attain the upper (lower) bounds.” First we investigate the structuralproperties of the extremal graphs. Then these results are used to characterize all extremal graphswith the largest signless Laplacian spectral radius in the set of tree and unicyclic graphic degreesequence, respectively. Consequently, we also obtain all extremal trees and unicyclic with the largestsignless Laplacian spectral radius in the sets of all trees and unicyclic graphs of order n with thelargest degree, the leaves number and the matching number, respectively.

Keywords: Signless Laplacian matrix; Degree sequence.

Sept. 18, 14:30 – 15:00 Tetsuji Taniguchi

On Fat Hoffman Graphs with Smallest Eigenvalue At Least −3

Hye Jin JangPohang University of Science and Technology

Jack KoolenPohang University of Science and Technology

Akihiro MunemasaTohoku University

Tetsuji TaniguchiMatsue College of Technology

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Email: tetsuzit@matsue-ct.ac.jp

Hoffman graphs are a limiting object of graphs with respect to the smallest eigenvalue. To understandgraphs with smallest eigenvalue −3, we investigate fat Hoffman graphs with smallest eigenvalue atleast −3, using their special graphs. We show that the special graph S(H) of an indecomposable fatHoffman graph H is represented by the standard lattice or a root lattice. Moreover, we show thatif the special graph admits an integral representation, that is, the lattice spanned by it is not anexceptional root lattice, then the special graph S−(H) is isomorphic to one of the Dynkin graphsAn, Dn, or extended Dynkin graphs An or Dn.

Sept. 18, 15:10 – 15:40 Guofu Yu

Blaszak-Marciniak Lattice, Toda Lattice and Combinatorial Numbers

Xiangke ChangChinese Academy of Sciences

Email: changxk@lsec.cc.ac.cn

Xingbiao HuChinese Academy of SciencesEmail: hxb@lsec.cc.ac.cn

Guofu YuShanghai Jiaotong UniversityEmail: gfyu@sjtu.edu.cn

Hankel type determinant solutions for the so-called generalized Blaszak-Marciniak lattice and (2+1)-dimensional Toda molecule equation are presented. As an application, the relations between thesesolutions and combinatorial numbers are discussed. Some new combinatorial numbers are obtained.

Keywords: Block Hankel determinant; Combinatorial numbers.

Sept. 18, 15:40 – 16:10 Stefan Grunewald

On Agreement Forests

Yang DingBoston College

Email: yding1985@yahoo.com.cn

Stefan GrunewaldCAS-MPG Partner Institute for Computational Biology

Email: stefan@picb.ac.cn

Peter J. HumphriesUniversity of Canterbury

Email: pjhumphries@gmail.com

The evolutionary relationship within a set of taxonomic units is commonly described by a leaf-labelled tree. Using different methods or different data sets frequently results in different trees

28

with the same label set. Further, some heuristic optimization algorithms require the constructionof a different but not too different tree from a given one, in order to find a (locally) optimal tree.Therefore, we have to quantify the dissimilarity between two phylogenetic trees with identical leafsets. Tree rearrangement operations are widely used to measure this dissimilarity. The tree bisectionand reconnection (tbr) distance for unrooted trees can be equivalently defined in terms of agreementforests. For both the tbr distance and the less general subtree prune and regraft (spr) distance, weuse such forests to derive new upper and lower bounds on the maximal possible distance betweentwo trees with n leaves.

Sept. 18, 16:20 – 16:50 Yoshio Sano

The Competition Number of a Graph and the Holes in the Graph

Yoshio SanoNational Institute of Informatics, Japan

Email: sano@nii.ac.jp

In this talk, I will give a survey of recent results on the relationship between the competition numberof a graph and the number of the holes in the graph.

Let D be an acyclic digraph. The competition graph of D, denoted by C(D), is the (simpleundirected) graph which has the same vertex set as D and has an edge between two distinct verticesx and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. Forany graph G, G together with sufficiently many isolated vertices is the competition graph of anacyclic digraph. From this observation, in 1978, Roberts [14] defined the competition number k(G)of a graph G to be the smallest number k such that G together with k isolated vertices is thecompetition graph of an acyclic digraph.

The notion of competition graph was introduced in 1968 by Cohen [2] in connection with a prob-lem in ecology. Since then, competition graphs and their variants have been defined and studied bymany authors. Since the characterization of competition graphs is equivalent to the computation ofcompetition numbers, it has been one of the important research problems in the study of competitiongraphs to compute the exact values of competition numbers. However it does not seem to be easyin general to compute k(G) for a given graph G, as Opsut [13] showed in 1982 that the computationof the competition number of a graph is an NP-hard problem.

A cycle in a graph is called an induced cycle (also called a chordless cycle or a simple cycle) ifit is an induced subgraph of the graph. A hole in a graph is an induced cycle of length at least 4 inthe graph. A graph without holes is called a chordal graph.

The competition number of a graph with a few holes has been studied: In 1978 Roberts [14]showed that the competition number of a chordal graph is at most 1; In 2005 Cho and Kim [1]showed that the competition number of a graph with exactly one hole is at most 2; In 2010 Lee,Kim, Kim, and Sano [8] and Li and Chang [12] showed that the competition number of a graph withexactly two holes is at most 3.

Recently, the relationship between the competition number and the number of holes of a graphwas studied more, and it has been shown that the competition number of a graph with exactly hholes is at most h + 1 under several assumptions by Li and Chang [11] in 2009, Kamibeppu [3] in2010, and Kim, Lee, and Sano [5] in 2010.

Furthermore, the above three results were generalized by Lee, Kim, and Sano [10] to the following:The competition number of a graph in which any two holes share at most one edge is at most thenumber of holes plus one.

Keywords: Competition graph; Competition number; Hole.

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References

[1] H. H. Cho and S. -R. Kim: The competition number of a graph having exactly one hole, DiscreteMathematics 303 (2005) 32–41.

[2] J. E. Cohen: Interval graphs and food webs: a finding and a problem, Document 17696-PR,RAND Corporation, Santa Monica, CA (1968).

[3] A. Kamibeppu: An upper bound for the competition numbers of graphs, Discrete AppliedMathematics 158 (2010) 154–157.

[4] S. -R. Kim: Graphs with one hole and competition number one, Journal of the Korean Mathe-matical Society 42 (2005) 1251–1264.

[5] S. -R. Kim, J. Y. Lee, and Y. Sano: The competition number of a graph whose holes do notoverlap much, Discrete Applied Mathematics 158 (2010) 1456–1460.

[6] S. -R. Kim, J. Y. Lee, and Y. Sano: Holes and a chordal cut in a graph, Preprint,arXiv:1103.4341

[7] S. -R. Kim, J. Y. Lee, B. Park, and Y. Sano: The competition number of a graph and thedimension of its hole space, Preprint, arXiv:1103.1028

[8] J. Y. Lee, S. -R. Kim, S. -J. Kim, and Y. Sano: The competition number of a graph withexactly two holes, Ars Combinatoria 95 (2010) 45–54.

[9] J. Y. Lee, S. -R. Kim, S. -J. Kim, and Y. Sano: Graphs having many holes but with smallcompetition numbers, Applied Mathematics Letters 24 (2011) 1331–1335.

[10] J. Y. Lee, S. -R. Kim, and Y. Sano: The competition number of a graph in which any two holesshare at most one edge, Preprint, arXiv:1102.5718

[11] B. -J. Li and G. J. Chang: The competition number of a graph with exactly h holes, all ofwhich are independent, Discrete Applied Mathematics 157 (2009) 1337–1341.

[12] B. -J. Li and G. J. Chang: The competition number of a graph with exactly two holes, Journalof Combinatorial Optimization, DOI: 10.1007/s10878-010-9331-9

[13] R. J. Opsut: On the computation of the competition number of a graph, SIAM Journal onAlgebraic and Discrete Methods 3 (1982) 420–428.

[14] F. S. Roberts: Food webs, competition graphs, and the boxicity of ecological phase space,Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo,Mich., 1976) (1978) 477–490.

Sept. 18, 16:50 – 17:20 Jeongmi Park

The Niche Graphs of Interval Orders

Jeongmi ParkPusan National University

Email: jm1015@pusan.ac.kr

The niche graph of a digraph D is the (simple undirected) graph which has the same vertex setas D and has an edge between two distinct vertices x and y if and only if N+

D (x) ∩ N+D (y) 6= ∅ or

N−D (x) ∩ N−

D (y) 6= ∅, where N+D (x) (resp. N−

D (x)) is the set of out-neighbors (resp. in-neighbors)of x in D. A digraph D = (V, A) is called an interval order (resp. a unit interval order) if thereexists an assignment J : V → 2R of a (unit) closed real interval J(x) ⊂ R to each vertex x ∈ V suchthat (x, y) ∈ A if and only if maxJ(y) < minJ(x). S.-R. Kim and F.S. Roberts characterized the

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competition graphs of interval orders in 2002, and Y. Sano characterized the competition-commonenemy graphs of interval orders in 2010.

In this talk, we give characterizations of the niche graphs of interval orders and unit intervalorders.

Keywords: Niche graph; Competition graph; Interval order; Unit interval order.

Sept. 18, 17:30 – 17:40 Peng Li

A Four Sweep LBFS Algorithm for Recognizing Interval Graphs

Peng Li, Yaokun WuShanghai Jiao Tong University

Email: lidetiansjtu@sjtu.edu.cn(PL),ykwu@sjtu.edu.cn(YW)

A graph is an interval graph if it is the intersection graph of intervals on a line. It is interestingto pursue an easily implementable recognition algorithm of interval graphs via multi-sweep graphsearches. In their monumental work [1, 2], Corneil, Olariu and Stewart propose a six sweep LBFSalgorithm for recognizing interval graphs and prove its correctness. They believe that the fivesweep version of their algorithm can also do the job and suggest that the proof may be much morecomplicated.

We recently propose a four sweep LBFS algorithm for recognizing interval graphs and prove itscorrectness. In this short presentation, we will introduce our new algorithm via examples and talkabout relevant stories.

References

[1] D.G. Corneil, S. Olariu, L. Stewart, The ultimate interval graph recognition algorithm?(extended abstract), Proceedings of the Ninth Annual ACM-SIAM Symposium on DiscreteAlgorithms, ACM, New York, SIAM, Philadelphia, 1998, pp. 175–180.

[2] D.G. Corneil, S. Olariu, L. Stewart, The LBFS structure and recognition of intervalgraphs, SIAM Journal on Discrete Mathematics, 23 (2009), pp. 1905–1953.

Sept. 18, 17:40 – 17:50 Xiaomei Chen

On Invariants of Digraphs

Sheng ChenHarbin Institute of Technology

Email: schen@hit.edu.cn

Xiaomei ChenHarbin Institute of TechnologyEmail: xmchenhit@gmail.com

In this talk we will present some invariants for digraphs. First we introduce the concept of dualtension on a digraph, show that the number of nonidentity dual tensions is a polynomial on the order

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of the finite group and discuss its relationships with the chromatic polynomial. Then consideringfunctional digraph as a basic structure, we present a two-variable polynomial for digraphs. Finallywe introduce a new invariant for digraphs under state in-splitting (and thus under line digraph)operation.

Keywords: Invariants; Digraphs; Dual-tension; Functional digraphs; In-splitting; Line digraphs.

Sept. 18, 17:50 – 18:00 Yangjing Long

Relations Between Graphs

Yangjing LongMax Planck Institute for Mathematics in the Sciences

Email: ylong@mis.mpg.de

Given two graphs G = (V (G), E(G)) and H = (V (H), E(H)), we ask under which conditions thereis a relation R ⊆ V (G) × V (H) that generates the edges of H given the structure of G. Thisconstruction generalizes full homomorphisms of graphs and naturally leads to generalized notions ofretractions, cores, and co-cores of graphs.

This is a joint work with Ling Yang, Peter F. Stadler and Jurgen Jost.

Sept. 18, 18:00 – 18:30 Simone Severini

A Role for the Lovasz theta Function in Quantum Mechanics

Simone SeveriniUniversity College London

Email: simoseve@gmail.com

The mathematical study of the transmission of information without error was initiated by Shannonin the 50s. Only in 1979, Lovasz solved the major open problem of Shannon concerned with thistopic. The solution is based on a now well-known object called Lovasz theta function. Its role greatlycontributed to the developments of areas of Mathematics like semidefinite programming and extremalproblems in combinatorics. The Lovasz theta function is an upper bound to the zero-error capacity,however it is not always tight. In the last two decades quantum information theory establisheditself as the natural extension of information theory to the quantum regime, i.e., for the study ofinformation storage and transmission with the use of quantum systems and dynamics. I will showthat the Lovasz theta function is an upper bound to the zero-error capacity when the parties using achannel can share certain quantum physical resources. This quantity, which is called entanglement-assisted zero-error capacity, can be greater than the classical zero-error capacity, in both, singleuse of the channel and asymptotically. Additionally, I will propose a physical interpretation of theLovasz theta function as the maximum violation of certain noncontextual inequalities. These areinequalities traditionally used to study the difference between classical, quantum mechanics, andmore exotic theories of nature. This framework allows to prove easily a number of known resultsand it gives a novel perspective in quantum information and combinatorial optimization.

Arranged on Sept. 19 or cancelled Yan Wang

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Self-dual and Self-Petrie-dual Regular Maps

R. Bruce RichterUniversity of Waterloo, Canada

Jozef SiranOpen University, U.K., and Slovak University of Technology, Slovakia

Yan WangYantai University, China

Email: yanwangmath@yahoo.com.cn

Regular maps are cellular decompositions of surfaces with the ‘highest level of symmetry’, notnecessarily orientation-preserving. Such maps can be identified with three-generator presentationsof groups G of the form G = 〈a, b, c| a2 = b2 = c2 = (bc)2 = (ab)n = (ca)m = . . . = 1〉; the positiveintegers m and n are the vertex degree and the face length of the map. A regular map (G; a, b, c)is self-dual if the assignment a 7→ a, b 7→ c and c 7→ b extends to an automorphism of G, andself-Petrie-dual if G admits an automorphism fixing a and c and interchanging b with bc.

In this note we show that for infinitely many numbers m there exist finite, self-dual and self-Petrie-dual regular maps of degree and face length equal to m. We also prove that no such mapwith odd vertex degree is a normal Cayley map.

Keywords: Regular map; Triangle group; Self-dual map; Self-Petrie-dual map.

Arranged on Sept. 19 or cancelled Taoyang Wu

On the TBR Graph of Tree Space

Taoyang WuNational University of SingaporeEmail: taoyang.wu@gmail.com

Phylogenetic tree space, the collection of all possible trees for a set of taxa, grows exponentiallywith the number of taxa, creating many computational challenges for tree inference. One approachto address it is by heuristic algorithms, which are typically based on a graph structure imposed onthe tree space: vertices are all possible trees with the same set of taxa, and two trees are adjacentif and only if one can be converted to the other by a local change. Therefore, understanding thesegraphs is important for designing algorithms, and interpreting the obtained results.

Among all popular local changes, tree bisection and reconnection (TBR) rearrangement is themost general one. In this talk we consider the graph induced by this rearrangement, and discusssome of its properties, including its diameter and the degree distribution.

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ang

Fuda

nU

nive

rsity

yanglingfd@fudan.edu.cn

Shun

ichi

Yok

oyam

aK

yush

uU

nive

rsity

s-yokoyama@math.kyushu-u.ac.jp

Guo

fuY

uSh

angh

aiJi

aoTon

gU

nive

rsity

gfyu@sjtu.edu.cn

35

Nam

eU

niv

ersi

tyEm

ail

Hou

yiY

uSh

angh

aiJi

aoTon

gU

nive

rsity

yhy178@163.com

Hyo

nJu

Yu

Poh

ang

Uni

vers

ity

ofSc

ienc

ean

dTec

hnol

ogy

lojs4110@postech.ac.kr

Wei

-Gan

gY

uan

Shan

ghai

Jiao

Ton

gU

nive

rsity

yuanweigang2005@yahoo.com.cn

Aix

ian

Zha

ngC

apit

alN

orm

alU

nive

rsity

zhangaixian1008@126.com

Gua

ng-J

unZha

ngSh

angh

aiJi

aoTon

gU

nive

rsity

guangjunzhang@126.com

Hai

-Jun

Zha

ngSh

angh

aiJi

aoTon

gU

nive

rsity

duncancancan@163.com

Jie

Zha

ngSh

angh

aiJi

aoTon

gU

nive

rsity

zhangjie.sjtu@163.com

Xia

o-D

ong

Zha

ngSh

angh

aiJi

aoTon

gU

nive

rsity

xiaodong@sjtu.edu.cn

Xiu

-Mei

Zha

ngSh

angh

aiJi

aoTon

gU

nive

rsity

wfluckyzxm@163.com

Yue

-Hui

Zha

ngSh

angh

aiJi

aoTon

gU

nive

rsity

zyh@sjtu.edu.cn

36

Index

Eiichi Bannai, 18Koichi Betsumiya, 16

Beifang Chen, 19Xiaomei Chen, 31Yuqun Chen, 8

Xavier Dahan, 25Yun-Ping Deng, 24

Keqin Feng, 13Rongquan Feng, 20Tao Feng, 22

Stefan Grunewald, 28Jun Guo, 23

Akihiro Higashitani, 20Mitsugu Hirasaka, 7

Takuya Ikuta, 15Tatsuro Ito, 13Yoshiaki Itoh, 16

Cuipo Jiang, 12

Teruhisa Kadokami, 20Kyoung-Tark Kim, 24Jacobus Koolen, 25Hirotake Kurihara, 26

Peng Li, 31Yangjing Long, 32

Jianmin Ma, 6Jun Ma, 19Prabhu Manyem, 23William J. Martin, 6Yoshihiro Mizoguchi, 22Akihiro Munemasa, 7

Nobuo Nakagawa, 11Chie Nara, 21Hiroshi Nozaki, 14

Takayuki Okuda, 18

Jeong Rye Park, 10Jeongmi Park, 30Jongyook Park, 26

Yoshio Sano, 29Simone Severini, 32Hao Shen, 14Yi-Huang Shen, 12Sho Suda, 7

Tetsuji Taniguchi, 27

Kaishun Wang, 26

Wei Wang, 10Yan Wang, 32Taoyang Wu, 33Yaokun Wu, 23

Ziqing Xiang, 18

Shun’ichi Yokoyama, 17Guofu Yu, 28Houyi Yu, 17Hyonju Yu, 15

Aixian Zhang, 13Xiao-Dong Zhang, 27

37

38

The above is the west corner of SJTU Minhang Campus in which the main gate, the Academic Activity Center and the Department of Mathematics are circled and marked with stars. For a complete Campus Map of SJTU, please visit: http://3dcampus.sjtu.edu.cn/en/index.html

License: Creative Commons Attribution-Noncommercial-Share Alikehttp://creativecommons.org/licenses/by-nc-sa/3.0/Please feel free to print and share.If you want to use this map commercially, contact us: exploremetro.com/contact

Shanghai Metro MapTo check ticket prices, find the fastest route, check train times, hear station names in Mandarin, and more, visit exploreshanghai.com

Updated May 2011

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1

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2

2

3

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3

4

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7

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6

6

10

10

10

8

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9

Xinzhuang – Fujin Road

East Xujing – Pudong International Airport

Shanghai South Railway Station – North Jiangyang Road

Loop line

Xinzhuang – Minhang Development Zone

Gangcheng Road – Oriental Sports Center

Shiguang Road – Aerospace Museum

Songjiang Xincheng – Middle Yanggao Road

Huamu Road – Shanghai University

Hangzhong Road/Hongqiao Railway Station– Xinjiangwancheng

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11

11

Jiangsu Road – Anting/North Jiading

Fujin Rd.

West Youyi Rd.

Bao’an Highway

Gongfu Xincun

Hulan Rd.

Tonghe Xincun

Gongkang Rd.

Penpu Xincun

Wenshui Rd.

Shanghai Circus World

Yanchang Rd.

North Xizang Rd.

Zhongxing Rd.North Zhongshan Rd.

Caoyang Rd.

Jiangsu Rd.

Jing’an Temple

Changshu Rd.

South Shaanxi Rd.

People’s SquareZhongshan Park

West Nanjing Rd.Xinzha Rd.

Hanzhong Rd.

Zhongtan Rd.

Baoshan Rd.

Hengshan Rd.Xujiahui

ShanghaiStadium

Laoximen

Shangcheng Rd.

South Huangpi Rd.

ShanghaiLibrary

JiantongUniversity

Dashijie

Xintiandi

Pudian Rd.

Damuqiao Rd. South Xizang Rd.

Lancun Rd.

Shanghai Indoor Stadium

Zhenping Rd.

Songhong Rd.

East Xujing

Beixinjing

Weining Rd.

Jinshajiang Rd.

Loushanguan Rd.

Hongqiao Rd.

West Yan’an Rd.

Zhongchun Rd.

Qibao

Hechuan Rd.

Yishan Rd.

SongjiangXincheng

Songjiang University Town

Dongjing Sheshan Sijing

Jiuting

Xingzhong Rd.

Guilin Rd.

Yili Rd.

Songyuan Rd.Shuicheng Rd.

Longxi Rd.

Shanghai Zoo

Hongqiao Airport T1

Longbai Xincun

Ziteng Rd.

Hangzhong Rd. Caobao Rd.

CaohejingHi-Tech Park

Minhang Development Zone

Wenjing Rd.

Huaning Rd.

Jinping Rd.

Dongchuan Rd.

Jianchuan Rd.

Beiqiao

Zhuanqiao

Yindu Rd.

Chunshen Rd.

Waihuan Rd.

Lianhua Rd.

Jinjiang Park

Shilong Rd.

Longcao Rd.

Caoxi Rd.

Luban Rd.

NanpuBridge

TangqiaoShanghai Children’sMedical Center

Shanghai South Railway Station

Oriental Sports Center

Xinzhuang

Dongming Rd.

Gaoqing Rd.

West Huaxia Rd.

Shangnan Rd.

South Lingyan Rd.

Linyi Xincun

Zhangjiang Hi-Tech Park

Century Park

Shanghai Science & Technology Museum

Lingkong Rd.YuandongAvenue

HaitiansanRoad

PudongInternational

Airport

Yuanshen Stadium

Minsheng Rd.

Beiyangjing Rd.

Deping Rd.

Yunshan Rd.

Jinqiao Rd.

Boxing Rd.

Wulian Rd.

Jufeng Rd.

Dongjing Rd.

Wuzhou Avenue

Zhouhai Rd.

South Waigaoqiao Free Trade Zone

Hangjin Rd.

North Waigaoqiao Free Trade Zone

Gangcheng Rd.Shiguang Rd.

Nenjiang Rd.

Xiangyin Rd.

Huangxing Park

Middle Yanji Rd.

Huangxing Rd.

Jiangpu Rd.

Anshan Xincun

Youdian Xincun

NorthSichuan Rd.

MiddleYanggao Rd.

YuyuanGarden

Tiantong Rd.

Siping Rd.Quyang Rd.Chifeng Rd.

Dabaishu

Jiangwan Town

West Yingao Rd.

South Changjiang Rd.

Songfa Rd.

Zhanghuabang

Songbin Rd.

Shuichan Rd.

Xinjiangwancheng

East Yingao Rd.

Sanmen Rd.

Jiangwan Stadium

Wujiaochang

Guoquan Rd.

Tongji University

Baoyang Rd.

Youyi Rd.

Tieli Rd.

North Jiangyang Rd.

Dongbaoxing Rd.

Hailun Rd.

Linping Rd.

Dalian Rd.

Yangshupu Rd.

LujiazuiEast Nanjing Rd.

Qufu Rd.

Dongchang Rd.

Century Avenue

HongkouFootballStadium

PudongAvenue

ShanghaiRailway Station

Chengshan Rd.

Yangsi

Luheng Rd.

Pujiang TownJiangyue Rd.

Lianhang Rd.

Aerospace Museum

Malu

Jiading Xincheng

Baiyin Road

ShanghaiCircuit

ShanghaiAutomobile City

EastChangji RoadAnting

West Jiading

North Jiading

Fengqiao Rd.

Zhenru

Shanghai West Railway Station

HongqiaoAirport T2

HongqiaoRailwayStation

Liziyuan

Qilianshan Rd.

Wuwei Rd.

Taopu Xincun

Nanxiang

Changping Rd.

Changshou Rd.

Langao Rd.

Xincun Rd.

Dahuasan Rd.

Xingzhi Rd.

Dachang Town

Changzhong Rd .

Shangda Rd.

Nanchen Rd.

Shanghai University

Longde Rd.

Zhaojiabang Rd.

Dong’an Rd.

Jiashan Rd. Madang Rd.

Dapuqiao Lujiabang Rd.

Xiaonanmen

West Gaoke Rd.

Huamu Rd.

Fanghua Rd.

Jinke Rd.

Guanglan Rd.

Tangzhen

Middle Chuangxin Rd.

East Huaxia Rd.

Chuansha

Jinxiu Rd.

South Yanggao Rd.

Yuntai Rd.

Changqing Rd.Houtan

Chuanchang Rd.Yaohua Rd.

Longyang Rd.

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10

Linzhao Xincun