Properties of finite-difference analog of one-dimensional Laplace operator on the graph

The finite-difference analogs of one-dimensional Laplace operator on the graph-star and the graph with a cycle are considered. At the same time differential operator characteristic continuity at its reduction to the finite-difference analog is essential: the structure of an eigenvalue set is similar to the structure of a proper value set of a differential operator, completeness of eigenvectors in the finite-dimensional space remains, the finite-difference analog of Laplace operator remains symmetric and positive.