2021/12/22 by Shaun Fallat, H. Tracy Hall, Fallat, Shaun M. +5 · 2 citations
Computer Science · Materials Science · Mathematics · #05C50 #15A18 #15B35 #15B57 #58C15 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Phase-change materials and chalcogenides
paper · pdf · doi:10.48550/arxiv.2112.11712
openalex publication_date 2021/12/22 · openalex created_date 2022/05/05 · openalex updated_date 2026/08/01
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.