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 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.

[1]  Yves Colin de Verdière,et al.  On a new graph invariant and a criterion for planarity , 1991, Graph Structure Theory.

[2]  Kevin N. Vander Meulen,et al.  Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method , 2010 .

[3]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[4]  J. M. Masqué,et al.  Analysis and algebra on differentiable manifolds , 2009 .

[5]  V. Arnold.  ON MATRICES DEPENDING ON PARAMETERS , 1971 .

[6]  B. Shader,et al.  The nilpotent-centralizer method for spectrally arbitrary patterns , 2013 .

[7]  Yves Colin de Verdière,et al.  Sur un nouvel invariant des graphes et un critère de planarité , 1990, J. Comb. Theory, Ser. B.

[8]  Shaun M. Fallat,et al.  Generalizations of the Strong Arnold Property and the Minimum Number of Distinct Eigenvalues of a Graph , 2015, Electron. J. Comb..

[9]  Warren E. Ferguson,et al.  The construction of Jacobi and periodic Jacobi matrices with prescribed spectra , 1980 .

[10]  T. Laffey.  Extreme nonnegative matrices , 1998 .

[11]  Hein van der Holst,et al.  The inertia set of a signed graph , 2012 .

[12]  S. Parter.  On the Eigenvalues and Eigenvectors of a Class of Matrices , 1960 .

[13]  Shaun M. Fallat,et al.  The inverse eigenvalue problem of a graph: Multiplicities and minors , 2017, J. Comb. Theory, Ser. B.

[14]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[15]  D. Olesky,et al.  Spectrally arbitrary patterns , 2000 .

[16]  Raphael Loewy,et al.  The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample , 2009 .

[17]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[18]  Gerry Wiener.  Spectral multiplicity and splitting results for a class of qualitative matrices , 1984 .