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2007 Fiscal Year Final Research Report Summary

Analysis of the relationship between the geometric structure of graphs and the spectra of discrete Laplacian

Research Project

Project/Area Number 16540116
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionShowa University

Principal Investigator

HIGUCHI Yusuke  Showa University, College of Arts and Sciences, Lecturer (20286842)

Project Period (FY) 2004 – 2007
Keywordsgraph / spectral geometry / Laplacian / covering structure / random walk / cover time
Research Abstract

For finite or infinite graphs, there are many kinds of researches on the relationship between geometric and spectral properties. Some of them clarify the similarities finite (infinite) graphs and compact (non-compact) manifolds: others clarify the difference between them. The present research is mainly concerned with spectral and geometric properties for infinite graphs form the latter point of view. Our main results are as follows:(i)We give sufficient condition for an abelian covering graph to have full spectrum property, that is, Laplacian on it has the whole interval [0, 2] as its spectrum;(ii) We show how the spectra change under the para-line operation, which is a kinds of graph operation;(iii) We give an estimate of the upper bounds of Dirichlet forms and using this estimate together with an h-transform, we show the equivalent between the essentially bipartiteness and a kind of symmetry of spectra;(iv) We show that, for a finite graph including a certain kind of a family of cycles, the spectrum of the Laplacian on its homology universal covering graph has band structure and no eigenvalues.

  • Research Products

    (10 results)

All 2007 2006 2004 Other

All Journal Article (8 results) (of which Peer Reviewed: 4 results) Presentation (2 results)

  • [Journal Article] Non-bipartiteness of graphs and the upper bounds of Dirichlet forms2006

    • Author(s)
      Yu.HIGUCHI and T.SHIRAI
    • Journal Title

      Potential Anal. 25

      Pages: 259-268

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Non-bipartiteness of graphs and the upper bounds of Dirichlet forms2006

    • Author(s)
      Yusuke, HIGUCHI, Tomoyuki, SHIRAI
    • Journal Title

      Potential Anal 25

      Pages: 259-268

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some spectral and geometric properties for infinite graphs2004

    • Author(s)
      Yu.HIGUCHI and T.SHIRAI
    • Journal Title

      AMS Contemp.Math. 347

      Pages: 29-56

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Some spectral and geometric properties for infinite graphs2004

    • Author(s)
      Yusuke, HIGUCHI, Tomoyuki, SHIRAI
    • Journal Title

      AMS Contemp. Math 347

      Pages: 29-56

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Spectral structure of the Laplacian on a covering graph

    • Author(s)
      Yu.HIGUCHI and Y.NOMURA
    • Journal Title

      European J.Combin. (In press)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Non-separating 2-factor of an even regular graph

    • Author(s)
      Yu.HIGUCHI and Y.NOMURA
    • Journal Title

      Discrete Math. (In press)

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Spectral structure of the Laplacian on a covering graph

    • Author(s)
      Yusuke, HIGUCHI, Yuji, NOMURA
    • Journal Title

      European J. Combin (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Non-separating 2-factor of an even regular graph

    • Author(s)
      Yusuke, HIGUCHI, Yuji, NOMURA
    • Journal Title

      Discrete Math (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] グラフの幾何とスペクトル2007

    • Author(s)
      樋口 雄介
    • Organizer
      日本数学会秋季総合分科会 応用数学分科会特別講演
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-21
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Geometry and Analysis on Graphs2007

    • Author(s)
      Yusuke, HIGUCHI
    • Organizer
      Special Talks, Division of Applied Mathematics, MSJ meeting
    • Place of Presentation
      Tohoku University, Sendai, JAPAN
    • Year and Date
      2007-09-21
    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2010-02-04  

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