Global Analysis of the Spectrum of an Infinite Graph

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We have obtained the following results :(1) We estimated the first eigenvalue and sectional curvature of compact symmetric spaces, and gave counter example of Elworthy-Rosenberg.(2) We obtained the Faber-Krahn type estimate for the first eigenvalue of the Dirichlet eigenvalue problem for a coonected finite graph with boundary.(3) We obtained the Barta type estimate of the Dirichlet eigenvalue problem for a graph, and sharp estimate of the infimum of the spectrum of an infinite graph.(4) We introduce the notion of surgery, and gave its application.(5) We characterized homogeneous spaces admitting affine projectively flat connections.(6) We gave the first variational formula of a natural functional on the space ofWeyl structures.(7) Harmonic morphisms play important roll in differential geometry oh harmonic maps. We consider the discrete harmonic morphisms, and obtained their complete characterization.(8) We obtained theory of solutions of the inhomogeneous Yang-Mills equation.(9) We gave a characterization of harmonic morphism between two graphs.(10) We obtain a general estimation formula of the spectrum of the discrete Laplacian for an infinite graph. Our method is quite new using our new notion of the incresing degree and decreasing one relative to distance from a fixed vertex.