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

对于有限或无限图，几何性质与谱性质之间的关系有多种研究。其中一些阐明了有限（无限）图和紧（非紧）流形的相似性;另一些阐明了它们之间的区别。本文主要从后一种观点出发，研究无限图的谱性质和几何性质。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Laplacian具有整个区间[0，2]作为其谱，（ii）给出了谱在仿线运算（一种图运算）下的变化规律;（iii）我们给出了Dirichlet形式的上界的估计，并使用这个估计与h-变换，证明了谱的本质二分性与谱的一种对称性之间的等价性;（iv）我们证明了，对于包含某类圈族的有限图，Laplacian在其同调泛覆盖图上的谱具有带结构且无特征值。

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.