Hakata Worksho 2013 on "Combinatorics and its Application"

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.

**Organizers**

- Yoshihiro Mizoguchi (Kyushu University),
- Hayato Waki (Kyushu University),
- Mitsugu Hirasaka (Pusan National University),
- Tetsuji Taniguchi (Matsue College of Technology),
- Osamu Shimabukuro (Sojo University)
- Laboratory of Advanced Software in Mathematics, Institute of Mathematics for Industry, Kyushu University

《

**Program**》

9:30--10:05 Akihiro Munemasa (Tohoku University)

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

Abstract:

A complex Hadamard matrix is an n by n matrix with complex

*H*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.

10:10--10:45 Minwon Na (Tohoku University)

Title:TBA

11:00--11:35 Michael Dobbins (KAIST)

Title:TBA

11:40--12:15 Aleksandar Jurišić (University of Ljubljana)

Title:TBA

12:20--15:10 Poster Session 「Introduction to Mathematical Software」(in Japanese)

15:15--16:15 Xiao-Dong Zhang (Shanghai Jiao Tong University)

Title:The algebraic connectivity of graphs

Abstract:

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.

16:30--17:05 Katsuhiro Ota (Keio University)

Title:TBA

17:10--17:45 Yota Otachi (JAIST)

Title:The path-distance-width of hypercubes

Abstract:

The path-distance-width of a connected graph

*G*is the minimum integer

*ω*satisfying that there is a nonempty subset of

*S*⊆

*V*(

*G*)such that the number of the vertices with distance

*i*from

*S*is at most

*ω*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.

This seminar supported by Global COE Program "Education and Research Hub for Mathematics-for-Industry"