Reconstruction of the Sturm--Liouville Operator on a Graph from Subinterval Nodes

We consider inverse subinterval-nodal problems for the Sturm--Liouville operator on a star graph with mixed boundary conditions in pendant vertices and the standard matching conditions at the interior vertex. Can subinterval nodes recognize star-shaped quantum graphs? In this paper we give a positive answer. It is shown that the data of subinterval-nodes near the interior vertex can uniquely determine the potential on a graph and boundary conditions.