Let G be a connected locally finite simplicial graph with rk(π1(G))[ges ]2 and let T be the universal cover of G. Consider a π1(G)-invariant conformal density μ of dimension d on ∂T. The total mass function ϕμ of μ is defined on the set of vertices of G. Let |ϕμ| be its l2-norm. Let Ω be the geodesic flow space of G and mμ the invariant measure on Ω associated to μ. The main results of this paper are the following:

(i) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [les ]k. Then, if d>½log(k−1) and mμ(Ω)<∞, we have ϕμ∈l2(V) and

|ϕμ|2[les ]Cmμ(Ω),

with C=(e2d−1)/(e2d−k+1).

(ii) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [ges ]k. Then, if ϕμ∈l2(V), we have d>½log(k−1) and

Cmμ(Ω)[les ]|ϕμ|2.