  Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and
$m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and
completeness of the wave operators $ W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$
denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and
$I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$.
In particular, this entails a very general criterion for the absolutely
continuous spectra of $H_1$ and $H_2$ to be equal.
