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

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