Abstract
In this work it is proposed to study differential operators on a graph as an operator composed of differential operators on one-dimensional arcs and matrix operators on interior vertices of the graph. The work explores some questions concerning the theoretical side of ordinary differential equations with integro-differential conditions on stratified sets like graph.The attention will be paid to reconstruction of the domain of differential operator on directed graph. The reconstruction of the domain of differential operator means a simple specifying the boundary conditions from a known differential equations and its known eigenvalues. The paper studies the case of the second order differential equations with irregular boundary conditions on the vertices of directed graph. To achieve our goal we use the fact that finite set of eigenvalues serves as additional information for reconstruction of the domain of the differential operator on stratified set. The constructive algorithms for reconstructing the domain of definition of differential operator on directed graph are developed. All boundary functions from the spectral data are uniquely restored.