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The inverse eigenvalue problem of a graph studies the real symmetric matriceswhose off-diagonal pattern is prescribed by the adjacencies of the graph. Thestrong spectral property (SSP) is an important tool for this problem. This noteestablishes the bifurcation lemma, which states that if a spectrum can berealized by a matrix with the SSP for some graph, then all the nearby spectracan also be realized by matrices with the SSP for the same graph. The idea ofthe bifurcation lemma also works for other strong properties and for notnecessarily symmetric matrices. This is used to develop new techniques forverifying a spectrally arbitrary pattern or inertially arbitrary pattern. Thebifurcation lemma provides a unified theoretical foundation for several knownresults, such as the stable northeast lemma and the nilpotent-centralizermethod.

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