Spectral reconstruction number for graph K4
Oleksandr Averkin, Larisa Tymoshkevych · Mohyla Mathematical Journal · 2025
In this work, we introduce new formulations of inverse spectral problems for weighted graphs in which certain spectral data (namely, the spectra of selected induced subgraphs) uniquely determine the edge weights of the original graph. To quantify this, we define the spectral reconstruction number of a graph Srn(G) as the minimum number of spectra of induced subgraphs required to uniquely recover all edge weights of G.Motivated by their broad range of applications, inverse spectral problems for various classes of matrices have been actively studied in the literature. These problems typically involve recovering a matrix, or part of it, from the spectrum of the matrix itself or from the spectra of its submatrices.From a matrix-theoretic perspective, the problem concerns irreducible symmetric matrices with zero diagonal and nonnegative off-diagonal entries, which are adjacency matrices of connected edge-weighted graphs. Thus, the results obtained here offer new inverse spectral formulations for this class of matrices.The main contribution of this paper is the exact determination of the spectral reconstruction number for the complete graph on four vertices.