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Inverse nodal problems for differential pencils on a star-shaped graph

In this paper, we deal with the inverse nodal problems of reconstructing the second-order pencils of ordinary differential operators on a star-shaped graph. First, we derive the asymptotic expressions of large eigenvalues and regularized the trace formula of this operator. Second, we prove that a dense subset of nodal points uniquely determines the parameters of the boundary conditions and the potential functions on a graph. We also provide a constructive procedure for the solution of the inverse nodal problems.