Scattering the geometry of weighted graphs

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 ${\mathrm{\ell }}^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of ${\mathrm{\ell }}^2(X,m_1)$ with ${\mathrm{\ell }}^2(X,m_2)$. In particular, this entails a very general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal.