We consider inverse nodal problems for the Sturm–Liouville operators on the tree graphs. Can only dense nodes distinguish the tree graphs? In this paper it is shown that the data of dense-nodes uniquely determines the potential (up to a constant) on the tree graphs. This provides interesting results for an open question implied in the paper.

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In this work, we study some properties of weight matrices for the Dirac operator on a star graph, which are corresponding concepts to those of the Sturm-Liouville operator. There are two major ingredients. The first is devoted to the structure properties and asymptotic formulae of weight matrices. As an application, we construct a sequence of eigenfunctions and prove its Riesz basicity in the second ingredient. The methods employed rely on contour integration techniques and the theory of Riesz bases.