  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.
