Acta Mathematica Vietnamica

, Volume 43, Issue 4, pp 641–647 | Cite as

The Change in Multiplicity of an Eigenvalue of a Real Symmetric Matrix Resulting from the Changes in Edge Values Around a Classified Vertex in a Tree

  • Kenji ToyonagaEmail author


Given a real symmetric matrix A whose graph is a tree T and its eigenvalues, vertices in T can be classified in three categories, based upon the change in multiplicity of a particular eigenvalue, when the vertex is removed. We investigate the change in multiplicity of an eigenvalue based upon changes in edge values around a classified vertex. Then, we observe that a 2-Parter edge, a Parter edge, and a downer edge are located separately from each other in a tree, and there is a neutral edge between them. Especially, we show that the distance between a downer edge and a 2-Parter edge or a Parter edge is at least 2. We also consider some relations between a downer edge and the value on a downer vertex.


Edges Eigenvalues Graph Matrix entries Multiplicity Real symmetric matrix Tree 

Mathematics Subject Classification (2010)

15A18 05C50 15B57 13H15 05C05 


  1. 1.
    Horn, R., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)Google Scholar
  2. 2.
    Johnson, C.R., Leal-Duarte, A., Saiago, C.M.: The Parter-Wiener theorem: refinement and generalization. SIAM J. Matrix Anal. Appl. 25(2), 352–361 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Johnson, C.R., Leal-Duarte, A., Saiago, C.M.: The change in eigenvalue multiplicity associated with perturbation of a diagonal entry. Linear Multilinear Algebra 60(5), 525–532 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Johnson, C.R., McMichael, P.R.: The change in multiplicity of an eigenvalue of a Hermitian matrix associated with the removal of an edge from its graph. Discrete Math. 311(2–3), 166–170 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Toyonaga, K., Johnson, C.R.: The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix resulting from the change in an edge value. Spec. Matrices 5, 51–60 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Creative EngineeringNational Institute of Technology, Kitakyushu CollegeKitakyushuJapan

Personalised recommendations