  In this paper, we develop two approaches to investigation of inverse spectral
problems for a new class of nonlocal operators on metric graphs. The Laplace
differential operator is considered on a star-shaped graph with nonlocal
integral matching condition. This operator is adjoint to the
functional-differential operator with frozen argument at the central vertex of
the graph. We study the inverse problem that consists in the recovery of the
integral condition coefficients from the eigenvalues. We obtain the spectrum
characterization, reconstruction algorithms, and prove the uniqueness of the
inverse problem solution.
