Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex

A spectral problem occurring in description of small transverse vibrations of a star graph of Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these vertices are clamped. One vertex (the root) can move with damping in the direction orthogonal to the equilibrium position of the strings. We describe the spectrum of such spectral problem. The corresponding inverse problem lies in recovering the values of point masses and the lengths of the intervals between the masses using the spectrum and some other parameters. We propose conditions on a sequence of complex numbers and a collection of real numbers to be the spectrum of a problem we consider and the lengths of the edges, correspondingly.