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グラフの最小固有値とユークリッド空間内の等角直線族

Research Project

Project/Area Number 14F03903
Research Category

Grant-in-Aid for JSPS Fellows

Allocation TypeSingle-year Grants
Section外国
Research Field Clinical psychology
Research InstitutionTohoku University

Principal Investigator

宗政 昭弘  東北大学, 情報科学研究科, 教授 (50219862)

Co-Investigator(Kenkyū-buntansha) GREAVES GARY  東北大学, 情報科学研究科, 外国人特別研究員
GREAVES Gary  東北大学, 情報科学研究科, 外国人特別研究員
Project Period (FY) 2014-04-25 – 2016-03-31
Project Status Completed (Fiscal Year 2015)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2015: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2014: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsグラフ / 固有値 / 等角直線族 / 実対称行列 / 三重対角行列
Outline of Annual Research Achievements

代数的なアプローチにより、異なる固有値の数に制限をつけたグラフの分類問題に取り組んだ。特に、強正則グラフの一般化である、異なる固有値が3個しかないグラフの分類問題において、いくつかの成果が出た。正則グラフの場合はこの条件から自動的に強正則グラフになるが、正則でないが異なる固有値が3個しかないグラフは存在し、その分類問題にはわかっていないことが多い。この問題において、次数が2種類の二部グラフであるものについて分類結果を得た。また、等角直線族の存在問題の中から、正則グラフの存在問題に帰着できる場合を見いだし、実際に非存在を証明することができた。この問題に関しては、昨年度修正を終えた論文がようやく出版された。また、この論文で未だ未解決となっているひとつの場合について、グラフ理論の技術を使って解決する道筋を得た。特に、異なる固有値が4個しかない正則グラフの存在問題に帰着することができ、先行研究の成果を利用することができた。さらに、グラフのラプラス固有値のみたす不等式について、等号成立の条件を特徴付ける問題に取り組んだ。一般に等号成立の条件を完全に特徴付けることは困難であるが、いくつかの場合には完全な特徴付けが見つかり、また他のいくつかの場合には隣接行列を分割して得られる商行列の固有値の性質で特徴付けられることがわかった。これらグラフのラプラス固有値に関する研究成果をまとめた論文は執筆中である。

Research Progress Status

27年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

27年度が最終年度であるため、記入しない。

Report

(2 results)
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (16 results)

All 2016 2015 2014

All Journal Article (2 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 2 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (14 results) (of which Int'l Joint Research: 4 results,  Invited: 1 results)

  • [Journal Article] Equiangular lines in Euclidean spaces2016

    • Author(s)
      G. Greaves, J. Koolen, A. Munemasa, and F. Szollosi
    • Journal Title

      J. Combin. Theory Ser. A

      Volume: 138 Pages: 208-235

    • DOI

      10.1016/j.jcta.2015.09.008

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Biregular graphs with three eigenvalues2016

    • Author(s)
      X.-M. Cheng, A. L. Gavrilyuk, G. R. W. Greaves, and J. H. Koolen
    • Journal Title

      European J. Combin.

      Volume: - Pages: 57-80

    • DOI

      10.1016/j.ejc.2016.03.004

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
  • [Presentation] Block intersection for strongly regular graphs2016

    • Author(s)
      G. Greaves
    • Organizer
      Korea-Japan Workshop on Algebra and Combinatorics
    • Place of Presentation
      Pusan National University, Korea
    • Year and Date
      2016-01-29
    • Related Report
      2015 Annual Research Report
  • [Presentation] On the clique number of strongly regular graphs2015

    • Author(s)
      G. Greaves
    • Organizer
      The Fourth Workshop on Spectral Graph Theory and Related Topics
    • Place of Presentation
      筑波大学(つくば市)
    • Year and Date
      2015-11-21
    • Related Report
      2015 Annual Research Report
  • [Presentation] On the clique number of strongly regular graphs2015

    • Author(s)
      G. Greaves
    • Organizer
      International Workshop on Algebraic Combinatorics
    • Place of Presentation
      Zhejiang University, China
    • Year and Date
      2015-09-22
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Equiangular lines in Euclidean spaces2015

    • Author(s)
      G. Greaves
    • Organizer
      Systems of Lines: Applications of Algebraic Combinatorics
    • Place of Presentation
      Worcester Polytechnic Institute, U.S.A.
    • Year and Date
      2015-08-14
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Graphs with second largest eigenvalue at most 12015

    • Author(s)
      G. Greaves
    • Organizer
      CoCoA 2015
    • Place of Presentation
      Colorado State University, U.S.A.
    • Year and Date
      2015-07-23
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Graphs with three eigenvalues and second largest eigenvalue at most 12015

    • Author(s)
      G. Greaves
    • Organizer
      8th Slovenian Conference on Graph Theory
    • Place of Presentation
      Kranjska Gora, Slovenia
    • Year and Date
      2015-06-23
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Graphs with three eigenvalues2015

    • Author(s)
      G. Greaves
    • Organizer
      Workshop on Algebraic Combinatorics
    • Place of Presentation
      Tilburg University, The Netherlands
    • Year and Date
      2015-06-18
    • Related Report
      2015 Annual Research Report
  • [Presentation] Graphs with three eigenvalues and bounded second largest eigenvalue2015

    • Author(s)
      G. Greaves
    • Organizer
      The Third Workshop on Spectral Graph Theory and Related Topics
    • Place of Presentation
      広島工業大学
    • Year and Date
      2015-03-15
    • Related Report
      2014 Annual Research Report
  • [Presentation] Eigenvalues of symmetric matrices and tridiagonal matrices2015

    • Author(s)
      G. Greaves
    • Organizer
      Japan-Korea Workshop on Algebra and Combinatorics
    • Place of Presentation
      九州工業大学
    • Year and Date
      2015-01-29
    • Related Report
      2014 Annual Research Report
  • [Presentation] Graphs with precisely three distinct eigenvalues2014

    • Author(s)
      G. Greaves
    • Organizer
      有限群とその表現, 頂点作用素代数, 代数的組合せ論の研究
    • Place of Presentation
      京都大学
    • Year and Date
      2014-12-19
    • Related Report
      2014 Annual Research Report
  • [Presentation] Eigenvalues of integer symmetric matrices2014

    • Author(s)
      G. Greaves
    • Organizer
      Algebraic Combinatorics Workshop
    • Place of Presentation
      中国科学技術大学、中国
    • Year and Date
      2014-11-27
    • Related Report
      2014 Annual Research Report
  • [Presentation] Graphs with precisely three distinct eigenvalues2014

    • Author(s)
      G. Greaves
    • Organizer
      The Japanese Conference on Combinatorics and its Applications
    • Place of Presentation
      筑波大学
    • Year and Date
      2014-08-26
    • Related Report
      2014 Annual Research Report
  • [Presentation] Equiangular lines in Euclidean spaces2014

    • Author(s)
      G. Greaves
    • Organizer
      Modern Trends in Algebraic Graph Theory
    • Place of Presentation
      Villanova University, U.S.A.
    • Year and Date
      2014-06-03
    • Related Report
      2014 Annual Research Report
  • [Presentation] Seidel matrices with precisely three distinct eigenvalues2014

    • Author(s)
      G. Greaves
    • Organizer
      Japan Conference on Graph Theory and Combinatorics
    • Place of Presentation
      日本大学
    • Year and Date
      2014-05-20
    • Related Report
      2014 Annual Research Report

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Published: 2015-01-22   Modified: 2024-03-26  

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