  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.
