On multiplicity of eigenvalues in quantum graph theory

Spectral problems are considered generated by the Sturm-Liouville equation on equilateral trees with the Dirichlet boundary conditions at pendant vertices and continuity and Kirchhoff's conditions at interior vertices. It is shown that the eigenvalues of such problems approach asymptotically the eigenvalues of the problem on the same tree with zero potentials on the edges. It is shown that between any two eigenvalues of maximal multiplicity (p(pen)-1) where p(pen) is the number of pendant vertices there are p(in) eigenvalues (with account of multiplicity, where p(in) is the number of interior vertices in the tree). (C) 2019 Published by Elsevier Inc.