The bifurcation lemma for strong properties in the inverse eigenvalue problem of a graph

TL;DR

This paper introduces the bifurcation lemma, a key theoretical tool that ensures the stability of the strong spectral property in inverse eigenvalue problems of graphs, enabling new verification techniques.

Contribution

The paper establishes the bifurcation lemma for the SSP, providing a unified foundation and extending its application to other strong properties and matrix classes.

Findings

Bifurcation lemma guarantees nearby spectra can also have SSP.

Develops new techniques for verifying spectrally arbitrary patterns.

Unifies several known results like the northeast lemma and nilpotent-centralizer method.

Abstract

The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of the graph. The strong spectral property (SSP) is an important tool for this problem. This note establishes the bifurcation lemma, which states that if a spectrum can be realized by a matrix with the SSP for some graph, then all the nearby spectra can also be realized by matrices with the SSP for the same graph. The idea of the bifurcation lemma also works for other strong properties and for not necessarily symmetric matrices. This is used to develop new techniques for verifying a spectrally arbitrary pattern or inertially arbitrary pattern. The bifurcation lemma provides a unified theoretical foundation for several known results, such as the stable northeast lemma and the nilpotent-centralizer method.