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 in X^0, the vertex set) is a copy of the unit interval within the graph's one-skeleton X^1 and is assigned a positive conductance c(xy). A reference "Lebesgue" measure on X^1 is built up by using Lebesgue measure with total mass c(xy) on each edge [x,y]. There are three natural operators on X : the transition operator P acting on functions on X^0 (the reversible Markov chain associated with the conductances), the averaging operator A over spheres of radius 1 on X^1, and the Laplace operator on X^1 (with Kirchhoff conditions weighted by c(.) at the vertices). The relation between the l^2-spectrum of P and the H^2-spectrum of the Laplacian was described by Cattaneo (Mh. Math. 124, 1997). In this paper we describe the relation between the l^2-spectrum of P and the L^2-spectrum of A. 
Donald I. Cartwright and Wolfgang Woess