Higher eigenvalues and isoperimetric inequalities on Riemannian manifolds and graphs

5 Analysis on weighted graphs 23 5.1 Measures on graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Discrete Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3 Green’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.4 Integration versus Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.5 Eigenvalues of Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.6 Heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.7 Co-area formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Research supported in part by NSF Grant No. DMS 98-01446 Supported by EPSRC Fellowship B/94/AF/1782 Supported in part by NSF Grant No. DMS 95-04834

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