Gelfand’s inverse problem for the graph Laplacian
Emilia Blåsten
LUT University, Lahti, FinlandHiroshi Isozaki
University of Tsukuba, JapanMatti Lassas
University of Helsinki, FinlandJinpeng Lu
University of Helsinki, Finland
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Abstract
We study the discrete Gelfand'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: , where is called the “set of the boundary vertices” and is called the “set of the interior vertices.” We consider the case where the vertices in the set and the edges connecting them are unknown. Assume that we are given the set and the pairs , where are the eigenvalues of the graph Laplacian and are the values of the corresponding eigenfunctions at the vertices in . We show that the graph structure, namely the unknown vertices in 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 of cardinality contains two extreme points. A point is called an extreme point of if there exists a point such that is the unique nearest point in from with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.
Cite this article
Emilia Blåsten, Hiroshi Isozaki, Matti Lassas, Jinpeng Lu, Gelfand’s inverse problem for the graph Laplacian. J. Spectr. Theory 13 (2023), no. 1, pp. 1–45
DOI 10.4171/JST/455