The Gel'fand's inverse problem for the graph Laplacian

Abstract

We study the 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 ∪ 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 (λj, φj|B), where λj are the eigenvalues of the graph Laplacian and φj|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 ⊆ G of cardinality |S| 2 contains two extreme points. A point x ∈ S is called an extreme point of S if there exists a point z ∈ 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.

Inverse problems for discrete heat equations and random walks for a class of graphs

“…However, the inverse interior spectral problem can be reduced to the discrete Gel'fand's inverse boundary spectral problem that we have studied in [18]. The details of this reduction can be found in Section 2.3.…”

“…Our approach follows our recent work [18]. We assume that the graph structure is unknown but in a class of finite graphs satisfying the Two-Points Condition (with respect to accessible nodes).…”

“…The results are proved by relating to the inverse problems for heat equations and the inverse spectral problem. We mention that examples of finite graphs satisfying the Two-Points Condition include trees, subgraphs of periodic lattices and their perturbations (see [18,Section 1.3]).…”

“…In [18], we studied the discrete version of an inverse spectral problem, where we adopted a formulation analogous to the Gel'fand's inverse problem on manifolds with boundary. The Gel'fand's inverse problem [53] for partial differential equations has been a paradigm problem in the study of the mathematical inverse problems and imaging problems arising from applied sciences.…”

“…In the discrete Gel'fand's inverse problem studied in [18], the given subset of vertices (accessible nodes) is called the "boundary vertices", and one considers the equations that are only satisfied on vertices outside of the boundary vertices. The solutions on the boundary vertices are determined by given boundary conditions.…”