Analysis on the relation of the spectral structure of graphs to the cover times of random walks

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We first focused on a cover time of random walks on a finite graph, we could obtain some explicit formula for a special class of graphs, what are called “spider graphs.” However this analysis was very restricted．So we find we have to get more detailed information on spectra of discrete Laplacian in order to apply our method to a wider class of graphs. Thus we decided to study carefully the relation between spectra and geometry of graphs. We consequently obtain some results on spectral scattering problems, spectral resonance and the behaviour of quantum walks on graphs.