Abstract
Differential operators on graphs often arise in mathematics and different fields of science such as mechanics, physics, organic chemistry, nanotechnology, etc. In this paper the solutions of the Dirichlet problem for a differential operator on a star-shaped graph are deduced. And the differential operator with standard matching conditions in the internal vertices and the Dirichlet boundary conditions at boundary vertices are studied. Task is a model the oscillation of a simple system of several rods with an adjacent end. In work the formula of the Green function of the Dirichlet problem for the second order equation on directed graph is showed. Spectral analysis of differential operators on geometric graphs is the basic mathematical apparatus in solving modern problems of quantum mechanics.