A brief survey on the spectral geometry of a finite or infinite graph is given. After the adjacency matrix, discrete Laplacian and discrete Green's formula are introduced, the spectral geometry of finite graphs, particularly, estimation of the first positive eigenvalue in terms of the Cheeger constant, examples of isospectral or cospectral graphs and the Faber=Krahn type inequality are discussed. For infinite graphs, spectrum of the discrete Laplacian, the heat kernel and Green kernel are estimated. Finally, a relation between the finite element method for the Dirichlet boundary eigenvalue problem and the eigenvalue problem of the adjacency matrix for a graph is given.
Bulletin of the Japan Society for Industrial and Applied Mathematics 12 (1), 29-45, 2002
The Japan Society for Industrial and Applied Mathematics