On characteristic functions of equilateral regular star-trees

A spectral problem generated by the Sturm-Liouville equation on the edges of a regular equilateral finite star-tree with the Dirichlet boundary conditions at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices is considered. The potential in the Sturm-Liouville equations on the edges is the same on each edge and symmetric with respect to the edge midpoint. The structure of the function whose set of zeros coincides with the spectrum of such a problem (characteristic function) is described.