  A new method for solving inverse spectral problems on quantum star graphs is
proposed. The method is based on Neumann series of Bessel functions
representations for solutions of Sturm-Liouville equations. The representations
admit estimates for the series remainders which are independent of the real
part of the square root of the spectral parameter. This feature makes them
especially useful for solving direct and inverse spectral problems requiring
calculation of solutions on large intervals in the spectral parameter.
Moreover, the first coefficient of the representation is sufficient for the
recovery of the potential. The method for solving the inverse spectral problem
on the graph consists in reducing the problem to a two-spectra inverse
Sturm-Liouville problem on each edge. Then a system of linear algebraic
equations is derived for computing the first coefficient of the series
representation for the solution on each edge and hence for recovering the
potential. The proposed method leads to an efficient numerical algorithm that
is illustrated by a number of numerical tests.
