The kernel of the Laplace–Beltrami operator on a decorated graph

For the Laplace–Beltrami operator (the operator is given by a Lagrangian plane $\Lambda$ ), an isomorphism between the its kernel and intersection of $\Lambda$ and fixed lagrangian plane is described. For the $\Delta^0$ operator with “continuity” conditions (on a connected finite graph with $n$ edges and $v$ vertices), the inequality $\dim$ ker $\Delta^0 \le n - v + 2$ is obtained. It is also proved that the quantity $n - v + 1 - \dim$ ker $\Delta^0$ cannot be reduce while adding new edges and manifolds.