On Discrete Spectrum of a Model Graph with Loop and Small Edges

We consider a perturbed graph consisting of two infinite edges, a loop, and a glued arbitrary finite graph γe with small edges, where γe is obtained by e−1 times contraction of some fixed graph and e is a small parameter. On the perturbed graph, we consider the Schrodinger operator whose potential on small edges can singularly depend on e with the Kirchhoff condition at internal vertices and the Dirichlet or Neumann condition at the boundary vertices. For the perturbed eigenvalue and the corresponding eigenfunction we prove the holomorphy with respect to e and propose a recurrent algorithm for finding all coefficients of their Taylor series.