The Green's function formalism is extended here to multi-point posed boundary-value problems of a special type occurring in some situations in applied mechanics. Problems which reduce to special systems of linear ordinary differential equations are considered. These are formulated on finite weighted graphs in such a way that every equation in the system governs a single unknown function and is defined on a single edge of the graph. The individual equations are put into a system format by means of contact and boundary conditions at the vertices and endpoints of the graph, respectively. Based on such a statement, the notion of the matrix of Green's type is introduced. Two methods are proposed for the analytic construction of such matrices. Illustrative examples from different areas of applied mechanics are presented.