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* Hakata Workshop 2013 [#z008da3b]
''～ Combinatrics and its Applications～''
&color(red){THIS IS A PROVISIONAL VERSION};
#br
This is a satellite seminar of [[the 11th Japan-Korea Workshop on Algebra and Combinatorics :http://comb.math.kyushu-u.ac.jp/?20130124]].
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 [#xb11a6e1]
-[[Yoshihiro Mizoguchi:http://imi.kyushu-u.ac.jp/~ym/]] (Kyushu University),~
-Hayato Waki (Kyushu University),~
-Mitsugu Hirasaka (Pusan National University),~
-[[Tetsuji Taniguchi:http://researchmap.jp/tetsuzit-14/]] (Matsue College of Technology),~
-Osamu Shimabukuro (Sojo University)
-[[Laboratory of Advanced Software in Mathematics, Institute of Mathematics for Industry, Kyushu University:http://www.imi.kyushu-u.ac.jp/eng/academic_staffs/department/34]]
**Supported by [#l7bc435b]
-[[Global COE Program "Education and Research Hub for Mathematics-for-Industry":http://gcoe-mi.jp/]]
**Date [#i35500dd]
-January 26, 2013. 9:30-18:00
**Location [#vcadf025]
-Seminar Room I (2F) in Reference Eki Higashi Building.
1-16-14 Hakata-Eki-Higashi, Hakata-Ku, Fukuoka City, 812-0013
(see http://www.re-rental.com/ , [[Google maps:http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=%E7%A6%8F%E5%B2%A1%E5%B8%82%E5%8D%9A%E5%A4%9A%E5%8C%BA%E5%8D%9A%E5%A4%9A%E9%A7%85%E6%9D%B11%E4%B8%81%E7%9B%AE16-14&aq=&sll=33.590188,130.425417&sspn=0.012888,0.021157&ie=UTF8&hq=&hnear=%E6%97%A5%E6%9C%AC,+%E7%A6%8F%E5%B2%A1%E7%9C%8C%E7%A6%8F%E5%B2%A1%E5%B8%82%E5%8D%9A%E5%A4%9A%E5%8C%BA%E5%8D%9A%E5%A4%9A%E9%A7%85%E6%9D%B1%EF%BC%91%E4%B8%81%E7%9B%AE%EF%BC%91%EF%BC%96%E2%88%92%EF%BC%91%EF%BC%94&ll=33.591064,130.424795&spn=0.012887,0.021157&z=16]] )
**Program [#h6d29d6d]
***January 26 (Saturday) at Room I in Reference Eki Higashi Building. [#s883cdcb]
-9:30--10:05 Akihiro Munemasa (Tohoku University)
--Complex Hadamard matrices contained in a Bose-Mesner algebra
-11:00--11:35 Michael Dobbins (GAIA, Postech)
--Reducing combinatorial to projective equivalence in realizability problems for polytopes.
-11:40--12:15 Aleksandar Jurišić (University of Ljubljana)
--TBA
--TOWER GRAPHS AND EXTENDED GENERALIZED QUADRANGLES
-12:20--15:10 Poster Session 「[[数学ソフトウェア紹介:http://comb.math.kyushu-u.ac.jp/?plugin=attach&pcmd=open&file=cfp_for_comb.pdf&refer=20130126]]」(Introduction to Mathematical Software) (and break)
--TBA
-15:15--16:15 Xiao-Dong Zhang (Shanghai Jiao Tong University)
--The algebraic connectivity of graphs
-(break)
-16:30--17:05 Katsuhiro Ota (Keio University)
--Clique minors, chromatic numbers for degree sequences
-17:10--17:45 Yota Otachi (JAIST)
--The path-distance-width of hypercubes
**Abstract [#sb7c7865]
***Akihiro Munemasa (Tohoku University) [#pa19cf13]
-Title:Complex Hadamard matrices contained in a Bose-Mesner algebra
-Abstract:
A complex Hadamard matrix &mimetex(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
&mimetex("(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) [#ze9edc62]
-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) [#q728cf69]
-Title:TOWER GRAPHS AND EXTENDED GENERALIZED QUADRANGLES
-Abstract:
Let &mimetex(\Gamma); be a complete multipartite graph with at least two parts
and each part of size at least &mimetex(2);. For example, &mimetex("K_{t\times n}");,i.e., the complement of &mimetex(t); copies of &mimetex(K_n);, and &mimetex("t,n \ge 2");. Then the local graphs and the &mimetex(\mu);-graphs (that is the graphs induced on the common neighbours of two vertices at distance &mimetex(2);) are again complete multipartite. They are actually the same graphs, in our example &mimetex("K_{(t-1)\times n}");. If &mimetex("t\ge 3");, then the geodesic closure ofany &mimetex(\mu);-graph is the graph we started with.
Let now &mimetex(\Gamma); be the point graph of a generalized quadrangle GQ&mimetex("(s,t)");. Then &mimetex(\Gamma); is strongly regular like the complete multipartite graph (its valency is &mimetex(s(t+1));, while the number of triangles on an edge in (a) &mimetex(\Gamma); is &mimetex(s-1);, and (b) the complement of &mimetex(\Gamma); is &mimetex(t+1);). Its &mimetex(\mu);-graphs are &mimetex("K_{1\times (t+1)}"); and when the generalized quadrangle is regular, the convex closures of &mimetex(\mu);-graphs are &mimetex("K_{2\times (t+1)}");.
We study a family of graphs, denoted by &mimetex({\cal F});, with the following
properties
(i) their &mimetex(\mu);-graphs are complete multipartite,
(ii) there exist adjacent vertices &mimetex(x);, &mimetex(y); and a vertex &mimetex(z); at distance &mimetex(2); from both &mimetex(x);, &mimetex(y); in &mimetex(\Gamma);, and the number &mimetex("\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 &mimetex({\cal F});.
***Xiao-Dong Zhang (Shanghai Jiao Tong University) [#wec50930]
-Title:The algebraic connectivity of graphs
-Abstract:
Let &mimetex(G); be a simple graph of order &mimetex(n); and
&mimetex( L(G)=D(G)-A(G) ); be its Laplacian matrix, where &mimetex( D(G)); and &mimetex( A(G) ); are the degree diagonal and adjacency matrices, respectively.
The the second smallest eigenvalue of &mimetex( L(G) ); is called the algebraic connectivity of &mimetex(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) [#mb37f86c]
-Title:Clique minors, chromatic numbers for degree sequences
-Abstract:
For a given graph &mimetex(G);, let &mimetex(\chi(G)); and &mimetex(h(G)); denote
the chromatic number, and the maximum size of clique minors of &mimetex(G);,
respectively.
The well-known Hadwiger's conjecture (1943) states that
&mimetex(\chi(G)\le h(G)); holds for every graph &mimetex(G);, which is wide open
for the graphs with &mimetex(\chi(G)\ge 7);.
In 2005, Robertson posed the "Hadwiger's conjecture for degree sequences."
For a graphical degree sequence &mimetex(D);, let &mimetex(\chi(D)); and &mimetex(h(D));
denote the maximum &mimetex(\chi(G)); and &mimetex(h(G));, respectively,
among the graphs &mimetex(G); having degree sequence &mimetex(D);.
Robertson's conjecture states that &mimetex(\chi(D)\le h(D)); for any degree
sequence &mimetex(D);.
This conjecture was recently confirmed by Dvořák and Mohar
by showing strongly that &mimetex(\chi(D)\le h'(D));, where
&mimetex(h'(D)); is the maximum size of topological clique minors of graphs
having degree sequence &mimetex(D);.
In this talk, we give an alternative and very short proof of
Robertson's conjecture.
Also, we shall discuss the values of &mimetex(\chi(D));, &mimetex(h(D)); and &mimetex(h'(D));
for some &mimetex(D);.
These results are based on a joint work with Guantao Chen and Ryo Hazama.
***Yota Otachi (JAIST) [#h17d65f1]
-Title: The path-distance-width of hypercubes
-Abstract:
The path-distance-width of a connected graph &mimetex(G); is the minimum
integer &mimetex(w); satisfying
that there is a nonempty subset of &mimetex(S \subseteq V(G)); such that the
number of the vertices
with distance &mimetex(i); from &mimetex(S); is at most &mimetex(w); for any nonnegative integer &mimetex(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.
