Essential self-adjointness of the graph-Laplacian

We study the operator theory associated with such infinite graphs G as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator Delta will depend not only on G but also on a prescribed positive real valued function c defined on the edges in G. In electrical network models, this function c will determine a conductance number for each edge. We show that the corresponding Laplace operator Delta is automatically essential self-adjoint. By this we mean that Delta is defined on the dense subspace D (of all the real valued functions on the set of vertices G(0) with finite support) in the Hilbert space l(2)(G(0)). The conclusion is that the closure of the operator Delta is self-adjoint in l(2)(G(0)), and so, in particular, that it has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line. We prove that generically our graph Laplace operator Delta=Delta(c) will have continuous spectrum. For a given infinite graph G with conductance function c, we set up a system of finite graphs with periodic boundary conditions such the finite spectra, for an ascending family of finite graphs, will have the Laplace operator for G as its limit. (C) 2008 American Institute of Physics.