# 20130126

Last-modified: 2013-02-06 (水) 13:41:06 (1751d)
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## Hakata Workshop 2013 †

～ Combinatrics and its Applications～

This is a satellite seminar of the 11th Japan-Korea Workshop on Algebra and Combinatorics . Our purpose of this meeting is giving an opportunity to make a speech and to commuticate with reserchers who study verious fields not only Combinatorics.

Further information is available from the organizers below.

### Date †

• January 26, 2013. 9:30-18:00

### List of Poster session speakers †

#### 「数学ソフトウェア紹介」(Introduction to Mathematical Software) †

1. 岩淵 勇樹 (面白法人カヤック), プログラム名: GraphiCalPad
2. 久保 浩平(九州大学理学部物理学科), プログラム名: 正規圧縮距離を用いたクラスタリング
3. 島袋 修(崇城大学 工学部), プログラム名: フュージョンスキームの探索
4. 野崎 寛(愛知教育大学), プログラム名: Magmaによる極大2距離集合の分類(Classification of maximal 2-distance sets by Magma)
5. 松下昂平(九州大学大学院数理学府), プログラム名: アフィン写像を用いた補間による2次元アニメーション作成ソフトウェア
6. 鹿間 章宏(大阪大学大学院 情報科学研究科), プログラム名: トーリックイデアルの二次生成判定法
7. 照本 直敏(九州大学数理学府), プログラム名: Q-det
8. 喜友名 朝也(九州大学大学院数理学府), プログラム名: PARI/GPによるE_8 lattice の部分集合の生成
9. 谷口哲至(松江高専数理科学科), プログラム名: スターコンプリメントテクニックと最小固有値が－２以上のグラフの生成
10. ダハン グザヴィエ(ＧＣＯＥ-ＭＩ), プログラム名: MAGMAによるRamanujan graphチェッカー
11. 古川 貴司(九州大学大学院数理学府), プログラム名: SSD Problem
12. 高妻倫太郎(立命館アジア太平洋大学), プログラム名: Truffe（トリュフ）
13. 秦攀(九州大学 マス・フォア・インダストリ研究所), プログラム名: 非線形時系列に対するモデル推定と選択のソフト

### Abstract †

#### Akihiro Munemasa (Tohoku University) †

• Title:Complex Hadamard matrices contained in a Bose-Mesner algebra
• Abstract:

A complex Hadamard matrix $H$ is an n by n matrix with complex entries with absolute value 1, such that rows are pairwise orthogonal with respect to the hermitian inner product. Recently, Ada Chan constructed a 15 by 15 complex Hadamard matrix using the adjacency matrix of the line graph of the Petersen graph. We found another such matrix, and then generalized it to an infinite family. In this talk, we focus on how to set up a system of polynomial equations for solving this kind of problem more efficiently than the naive approach. This is achieved by determining the ideal of the 3-dimensional algebraic variety consisting of the points of the form $(x+1/x,y+1/y,z+1/z,x/y+y/x,y/z+z/y,z/x+x/z)$ in the 6-dimensional space. This is a joint work with Takuya Ikuta.

#### Michael Dobbins (GAIA, Postech) †

• Title:Reducing combinatorial to projective equivalence in realizability problems for polytopes.
• Abstract:

Determining if there is a polytope of any combinatorial type that satisfies some given property is made difficult by the fact that there are polytopes with realization spaces that are homotopic to any primary semialgebraic set. I will show how, for certain properties, this can be reduced to finding such realizations among projective equivalence classes of polytopes, which are much nicer spaces. An application of this answers a question posed by Bernt Lindström in 1971, that there does exist a polytope without an antiprism.

#### Aleksandar Jurišić (University of Ljubljana) †

• Title:TOWER GRAPHS AND EXTENDED GENERALIZED QUADRANGLES
• Abstract:

Let $\Gamma$ be a complete multipartite graph with at least two parts and each part of size at least $2$ . For example, $K_{t\times n}$ ,i.e., the complement of $t$ copies of $K_n$ , and $t,n \ge 2$ . Then the local graphs and the $\mu$ -graphs (that is the graphs induced on the common neighbours of two vertices at distance $2$ ) are again complete multipartite. They are actually the same graphs, in our example $K_{(t-1)\times n}$ . If $t\ge 3$ , then the geodesic closure ofany $\mu$ -graph is the graph we started with. Let now $\Gamma$ be the point graph of a generalized quadrangle GQ $(s,t)$ . Then $\Gamma$ is strongly regular like the complete multipartite graph (its valency is $s(t+1)$ , while the number of triangles on an edge in (a) $\Gamma$ is $s-1$ , and (b) the complement of $\Gamma$ is $t+1$ ). Its $\mu$ -graphs are $K_{1\times (t+1)}$ and when the generalized quadrangle is regular, the convex closures of $\mu$ -graphs are $K_{2\times (t+1)}$ . We study a family of graphs, denoted by ${\cal F}$ , with the following properties

(i) their $\mu$ -graphs are complete multipartite,

(ii) there exist adjacent vertices $x$ , $y$ and a vertex $z$ at distance $2$ from both $x$ , $y$ in $\Gamma$ , and the number $\alpha_\G=\alpha(x,y,z)$ of common neighbours of these vertices does not depend on a choice of such a triple.

This is a generalization of the study of extended generalized quadrangles in it is intimately connected with even more general study of locally strongly regular graphs. We report on our progress of the classification of the family ${\cal F}$ .

#### Xiao-Dong Zhang (Shanghai Jiao Tong University) †

Let $G$ be a simple graph of order $n$ and $L(G)=D(G)-A(G)$ be its Laplacian matrix, where $D(G)$ and $A(G)$ are the degree diagonal and adjacency matrices, respectively. The the second smallest eigenvalue of $L(G)$ is called the algebraic connectivity of $G.$ In this talk, we survey some new results and progress on the algebraic connectivity. In particular, we present some relationships between the algebraic connectivity and the graph parameters, such as the clique number, the matching number, the independence number, the isoperimetric number, etc.

#### Katsuhiro Ota (Keio University) †

For a given graph $G$ , let $\chi(G)$ and $h(G)$ denote the chromatic number, and the maximum size of clique minors of $G$ , respectively. The well-known Hadwiger's conjecture (1943) states that $\chi(G)\le h(G)$ holds for every graph $G$ , which is wide open for the graphs with $\chi(G)\ge 7$ . In 2005, Robertson posed the "Hadwiger's conjecture for degree sequences." For a graphical degree sequence $D$ , let $\chi(D)$ and $h(D)$ denote the maximum $\chi(G)$ and $h(G)$ , respectively, among the graphs $G$ having degree sequence $D$ . Robertson's conjecture states that $\chi(D)\le h(D)$ for any degree sequence $D$ . This conjecture was recently confirmed by Dvořák and Mohar by showing strongly that $\chi(D)\le h'(D)$ , where $h'(D)$ is the maximum size of topological clique minors of graphs having degree sequence $D$ . In this talk, we give an alternative and very short proof of Robertson's conjecture. Also, we shall discuss the values of $\chi(D)$ , $h(D)$ and $h'(D)$ for some $D$ . These results are based on a joint work with Guantao Chen and Ryo Hazama.

#### Yota Otachi (JAIST) †

The path-distance-width of a connected graph $G$ is the minimum integer $w$ satisfying that there is a nonempty subset of $S \subseteq V(G)$ such that the number of the vertices with distance $i$ from $S$ is at most $w$ for any nonnegative integer $i$ . We present a general lower bound on the path-distance-width of graph, and determine the path-distance-width of hypercubes by using the lower bound. We also discuss the applicability of the lower bound to other graphs.