Inverse problems for the differential operator on a graph with cycles

This work deals with inverse problems of the Sturm–Liouville operator on a graph with multi-cycles, endowed with standard matching conditions in the internal vertex and with Neumann boundary conditions at the boundary vertices. We show that the potential on a graph with multi-cycles (including the potential on cycles) can be constructed by the dense nodal points on the interval considered. Moreover, we investigate the so-called incomplete inverse problems of recovering the potential on a fixed edge from a subset of nodal points situated only on a part of the edge.