An Agmon estimate for Schrödinger operators on graphs

The Agmon estimate shows that eigenfunctions of Schrödinger operators, -Δϕ+Vϕ=Eϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\Delta \phi + V \phi = E \phi $$\end{document}, decay exponentially in the ‘classically forbidden’ region where the potential exceeds the energy level x:V(x)>E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ x: V(x) > E \right\} $$\end{document}. Moreover, the size of |ϕ(x)|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\phi (x)|$$\end{document} is bounded in terms of a weighted (Agmon) distance between x and the allowed region. We derive such a statement on graphs when -Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta $$\end{document} is replaced by the graph Laplacian L=D-A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L = D-A$$\end{document}: we identify an explicit Agmon metric and prove a pointwise decay estimate in terms of the Agmon distance.