Eigenvalue asymptotics for a star-graph damped vibrations problem

We consider a boundary value problem generated by Sturm-Liouville equations on the edges of a star-shaped graph. Thereby a continuity condition and a condition depending on the spectral parameter is imposed at the interior vertex, corresponding to the case of damping in the problem of small transversal vibrations of a star graph of smooth inhomogeneous strings. At the pendant vertices Dirichlet boundary conditions are imposed. We describe the eigenvalue asymptotics of the problem under consideration. AMS Classification Numbers: Primary 34B45. Secondary 34B07, 34B24