Article,
Walk generating functions and spectral measures of infinite graphs
Linear Algebra and its Applications, (1988)
DOI: 10.1016/0024-3795(88)90245-5
Abstract
Let G be a locally finite graph with bounded vertex degrees. Its adjacency matrix A gives rise to a bounded self-adjoint operator on l2(V(G>)). Let E(λ) be the resolution of the identity for A, and ev the vector in l2(V(G)) having the v-coordinate 1 and all other coordinates 0. The function μuv(λ)=(E(λ)eu,ev) is called the spectral measure of G corresponding to vertices u,v of G. It is related to the generating function Wuv(z) for the number of walks of given length between u and v. The spectral measures are used to show the following. The pairing theorem characterizing finite bipartite graphs in terms of their spectrum is generalized from finite to infinite graphs. The average closed-walk generating function can be defined in some cases, generalizing the Plancherel measure to not necessarily vertex-transitive graphs. Some theorems about the convergence of graph measures are given, with applications to the calculation of the expected eigenvalue distribution of a large finite random regular or bipartite semiregular graph, and the calculation of spectral measures of several examples.
