Quadratic functionals and nondegeneracy of boundary value problems on a geometric graph

Quadratic functionals defined on the space of functions differentiable on a geometric graph are considered. Analogs of the Lagrange and Dubois–Raymond lemmas are proved. Necessary extremum conditions for these quadratic functionals are obtained. A boundary value problem with conditions posed locally at the vertices of a geometric graph is shown to be selfadjoint if and only if it is generated by a quadratic functional. A subclass of quadratic energy functionals is singled out. The space of solutions of the homogeneous boundary value problem generated by a quadratic energy functional is described, and nondegeneracy criteria for such boundary value problems are derived.