Inverse eigenvalue problem for discrete Schrödinger operators of a graph
Anzila Laikhuram, Jephian C. -H. Lin
TL;DR
The paper studies the inverse eigenvalue problem for discrete Schrödinger operators $0ar{S}(G)$, a subclass of generalized adjacency matrices with nonpositive off-diagonal entries, by leveraging the strong spectral property (SSP). It develops signed analogues of the classical supergraph, liberation, and bifurcation lemmas and uses them to construct matrices with prescribed spectra while controlling signs and multiplicities. The authors classify realizable spectra for all graphs with at most five vertices, showing that $0ar{S}(G)$ often restricts spectra relative to $S(G)$ and that SSP-based tools are essential for these constructions. The findings advance understanding of vibration-inspired inverse problems on graphs, provide concrete spectra realizability criteria for small graphs, and suggest directions for further exploration of cycles and signed SSP extensions.
Abstract
A discrete Schrödinger operator of a graph $G$ is a real symmetric matrix whose $i,j$-entry, $i \neq j$, is negative if $\{i,j\}$ is an edge and zero if it is not an edge, while diagonal entries can be any real numbers. The discrete Schrödinger operators have been used to study vibration theory and the Colin de Verdière parameter. The inverse eigenvalue problem for discrete Schrödinger operators of a graph aims to characterize the possible spectra among discrete Schrödinger operators of a graph. Compared to the inverse eigenvalue problem of a graph, the answers turn out to be more limited, and several restrictions based on graph structure are given. Using the strong properties, analogous versions of the supergraph lemma, the liberation lemma, and the bifurcation lemma are established. Using these results, the inverse eigenvalue problem for discrete Schrödinger operators is resolved for each graph with at most $5$ vertices.