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

TL;DR

This paper investigates the inverse eigenvalue problem for discrete Schr"odinger operators on graphs, providing characterizations and solutions for graphs with up to five vertices, with implications for vibration theory and graph parameters.

Contribution

It introduces new restrictions based on graph structure and solves the inverse eigenvalue problem for all graphs with up to five vertices.

Findings

Characterization of possible spectra for small graphs

Development of supergraph, liberation, and bifurcation lemmas

Complete solution for graphs with at most 5 vertices

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.