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* Hakata Workshop; Summer Meeting 2019 [#s7c20db8]
''～Discrete Mathematics and its Applications～''

#br
Our purpose of this meeting is giving an opportunity to make a speech and to communicate with researchers who study various fields not only Combinatorics.

Further information is available from the organizers below.

**Organizers [#ud21e362]

-[[Yoshihiro Mizoguchi:http://imi.kyushu-u.ac.jp/~ym/]] (Kyushu University),~
-[[Tetsuji Taniguchi:http://researchmap.jp/tetsuzit-14/]] (Hiroshima Institute of Technology),~
-[[Osamu Shimabukuro:http://research.jimu.nagasaki-u.ac.jp/IST?ISTActId=FINDJPDetail&ISTKidoKbn=&ISTErrorChkKbn=&ISTFormSetKbn=&ISTTokenChkKbn=&userId=100000912]] (Nagasaki University),~
-[[Makoto Tagami:https://sites.google.com/site/tagami77/]] (Kyushu Institute of Technology),
-[[Hirotake Kurihara:https://www.kct.ac.jp/seeds/ippann_rikei/kurihara.html]] (Kitakyushu National College of Technology),~
-[[Shuya Chiba:http://www.srik.kumamoto-u.ac.jp/chiba/chiba00.html]] (Kumamoto University),~
-[[Tsuyoshi Miezaki:https://sites.google.com/site/tmiezakij/]] (University of the Ryukyus),~

**Supported by [#i9925d50]
-[[Graduate School of Mathematics, Kyushu University:http://www.math.kyushu-u.ac.jp/eng/]]
-JSPS KAKENHI(Grant-in-Aid for Scientific Research (C)) Grant Number 17K05346
-JSPS KAKENHI(Grant-in-Aid for Scientific Research (C)) Grant Number 16K05263
-JSPS KAKENHI(Grant-in-Aid for Scientific Research (C)) Grant Number 19K03425

**Date [#b90d354b]
June 15, 2019

**Location [#g8525f22]
Seminar Room P (4F) 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 [#c1588fb3]
||~Speaker|~Title|
|15:07--15:10|>|Opening (Tetsuji Taniguchi)|
|15:10-15:50| Tetsuji Taniguchi (Hiroshima Institute of Technology)|[[A generalization of Hoffman graph >#g4zzad7i]]|
|16:00-16:40| Yagita Tsuyoshi (Kyushu Institute of Technology)|[[ On $k$-path Vertex Cover problem and its maximization >#bipnnfrm]]|
|16:50-17:30| Aokage Kazuya (National Institute of Technology, Ariake College)|[[Tensor products for the group related to the symmetric groups>#rh97xtyp]]|
|17:30--17:35|>|Closing|

**Abstract [#oc1154db]
*** Tetsuji Taniguchi [#g4zzad7i]
-Title: A generalization of Hoffman graph
-Abstract:
ホフマングラフはWoo氏，Neumaier氏らによって，最小固有値が$-1-\sqrt {2}$以上 ($-2$未満)であるグラフの研究の中で導入された。
与えられた値$\lambda (\le-2)$に対し，(従来の)ホフマングラフ$\ge\lambda$の和で表せないグラフが存在する。そこで，我々はfat頂点に重みを，辺に符号``$\pm$"を与えることでホフマングラフの一般化を与えた。これにより，すべてのグラフ$\ge\lambda$が(新しい)ホフマングラフの和で表すことが出来るようになった。
本講演ではこのことについて幾つかの結果を交えて話したい。
Hoffman graphs were introduced by Woo and Neumaier to study the graphs with smallest eigenvalue $\ge -1-\sqrt {2}$ but $<-2$. For a given value $\lambda (\le -2)$, there exist graphs with smallest eigenvalue $\ge\lambda$ that they can not be represented by a sum of
(usual) Hoffman graphs with smallest eigenvalue $\ge\lambda$. Therefore, we consider a further generalization of Hoffman graph by giving to the fat vertices a weight and by giving to the edges a sign ``$\pm$". By using this latest generalization, it becames possible to such graphs to be represented by a sum of (new) Hoffman graphs with smallest eigenvalue $\ge\lambda$.
In this talk we consider the above described generalization of Hoffman graphs and we give some results about them.


*** Yagita Tsuyoshi[#bipnnfrm]
-Title: On $k$-path Vertex Cover problem and its maximization
-Abstract:
Vertex Cover Problem (\texttt{VCP}) is one of the most popular graph optimization problems. Given a graph $G=(V,E)$ and integer $s$, \texttt{VCP} asks to find a vertex subset $S$ of size at most $s$ such that $G[V\setminus S]$ induces no edges. Recently, $k$-path Vertex Cover Problem (\texttt{$P_k$VCP}) was proposed and attracted much attention. \texttt{$P_k$VCP} aims to monitor all paths of $k$ vertices, thus is a natural generalization of the original vertex cover. In this talk, we talk about its maximization version and show the obtained results.

*** Aokage Kazuya [#rh97xtyp]
-Title: Tensor products for the group related to the symmetric groups
-Abstract:
The covering groups of the finite group G is introduced by Schur who investigated the projective representations of G.  We consider the
tensor products of the covering groups for the symmetric group. In this case, the tensor products have (i) linear and linear, (ii) linear
and spin, and (iii) spin and spin. When n is odd, Stembridge(1989) derived the results of (ii) and (iii)
for the basic spin. In this talk, we present the results when n is even.