Gel'fand's inverse problem for the graph Laplacian

We study the discrete Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=B\cup G$, where $B$ is called the set of the boundary vertices and $G$ is called the set of the interior vertices. We consider the case where the vertices in the set $G$ and the edges connecting them are unknown. Assume that we are given the set $B$ and the pairs $(\lambda_j,\phi_j|_B)$, where $\lambda_j$ are the eigenvalues of the graph Laplacian and $\phi_j|_B$ are the values of the corresponding eigenfunctions at the vertices in $B$. We show that the graph structure, namely the unknown vertices in $G$ and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset $S\subseteq G$ of cardinality $|S|\geqslant 2$ contains two extreme points. A point $x\in S$ is called an extreme point of $S$ if there exists a point $z\in B$ such that $x$ is the unique nearest point in $S$ from $z$ with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.