Inverse eigenvalue problem for discrete Schr\"odinger operators of a graph

Abstract

A discrete Schr\"odinger 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 Schr\"odinger operators have been used to study vibration theory and the Colin de Verdi\`ere parameter. The inverse eigenvalue problem for discrete Schr\"odinger operators of a graph aims to characterize the possible spectra among discrete Schr\"odinger 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 Schr\"odinger operators is resolved for each graph with at most $5$ vertices.