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

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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.