The Spectrum of the Averaging Operator on a Network (metric Graph)

A network is a countable, connected graph X viewed as a one-complex, where each edge [x, y] = [y, x] (x, y ∈ X, the vertex set) is a copy of the unit interval within the graph’s one-skeleton X and is assigned a positive conductance c(xy). A reference “Lebesgue” measure X is built up by using Lebesgue measure with total mass c(xy) on each edge xy. There are three natural operators on X: the transition operator P acting on functions on X (the reversible Markov chain associated with c), the averaging operator A over spheres of radius 1 on X, and the Laplace operator ∆ on X (with Kirchhoff conditions weighted by c at the vertices). The relation between the `-spectrum of P and the H-spectrum of ∆ was described by Cattaneo [4]. In this note we describe the relation between the `-spectrum of P and the L-spectrum of A.