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

图的几何结构与离散拉普拉斯谱的关系分析

• 批准号: 16540116
• 负责人: HIGUCHI Yusuke
• 金额: $ 2.46万
• 依托单位: Showa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540116/
• 关键词: graph; spectral geometry; Laplacian; covering structure; random walk; cover time; 状態密度関数

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.

对于有限或无限图，关于几何性质和谱性质之间关系的研究有很多种。其中一些阐明了有限（无限）图和紧致（非紧）流形的相似性：其他则阐明了它们之间的差异。目前的研究主要关注后一种观点的无限图的谱和几何性质。我们的主要结果如下：(i)给出了阿贝尔覆盖图具有全谱性质的充分条件，即其上的拉普拉斯算子具有整个区间[0, 2]作为其谱；(ii)我们展示了如何准线操作下的光谱变化，这是一种图操作；（iii）我们给出了狄利克雷形式的上限的估计，并将该估计与 h 变换一起使用，我们显示了本质上之间的等价二分性和一种谱的对称性；(iv)我们证明，对于包含某一类环族的有限图，其同调通用覆盖图上的拉普拉斯谱具有能带结构并且没有特征值。

项目成果

Some spectral and geometric properties for infinite graphs

无限图的一些谱和几何性质

• 发表时间: 2004
• 期刊: Contemp. Math. 347
• 作者: Yu.Higuchi;T.Shirai
• 通讯作者: T.Shirai

Non-separating 2-factor of an even regular graph

偶正则图的非分离二因子

• 期刊: Discrete Math. (In press)
• 作者: Yu;HIGUCHI;T;SHIRAI;Yu.HIGUCHI and Y.NOMURA;Yu.HIGUCHI and Y.NOMURA
• 通讯作者: Yu.HIGUCHI and Y.NOMURA

Spectral structure of the Laplacian on a covering graph

覆盖图上拉普拉斯算子的谱结构

• 发表时间: 2009
• 期刊: European Journal of Combinatorics
• 影响因子: 1
• 作者: J. Harada;S. Hashimoto and M. Otani;根上生也;Y.Takei;Hiroshi Kokubu;田村 英男;Tetsuji Tokihiro;前園宜彦;Y.Yamada;Y. Higuchi and Y. Nomura
• 通讯作者: Y. Higuchi and Y. Nomura