2007 Fiscal Year Final Research Report Summary
Analysis of the relationship between the geometric structure of graphs and the spectra of discrete Laplacian
| Project/Area Number |
16540116
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Showa University |
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Principal Investigator |
HIGUCHI Yusuke Showa University, College of Arts and Sciences, Lecturer (20286842)
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| Project Period (FY) |
2004 – 2007
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| Keywords | graph / spectral geometry / Laplacian / covering structure / random walk / cover time |
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| 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.
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Research Products
(10 results)
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[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
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[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
