Biharmonic functions on Schrödinger networks

In an infinite graph X,  the equation Δux=φxux,φx≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta u\left( x\right) =\varphi \left( x\right) u\left( x\right) ,\varphi \left( x\right) \ge 0,$$\end{document} is termed the Schrödinger equation. The solutions of this equation are named φ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi -$$\end{document}harmonic functions. This article studies varied aspects associated with these solutions: Harnack principle, minimum principle, domination principle, Dirichlet solution etc. The main thrust is the development of an abstract theory of discrete φ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi -$$\end{document}biharmonic functions on X and use it for the φ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi -$$\end{document}biharmonic classification of infinite graphs.