Convex and quasiconvex functions in metric graphs

<p style='text-indent:20px;'>We study convex and quasiconvex functions on a metric graph. Given set of points in the graph, we consider largest function below prescribed datum. We characterize this as unique viscosity subsolution to simple differential equation, <inline-formula><tex-math id="M1">\begin{document}$ u'' = 0 $\end{document}</tex-math></inline-formula> edges, plus nonlinear transmission conditions at vertices. also analogous problem for obtain characterization that is given datum.</p>