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

Abstract

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.

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Correspondence to Kenji Toyonaga.

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Toyonaga, K. 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. Acta Math Vietnam 43, 641–647 (2018). https://doi.org/10.1007/s40306-018-0276-x

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Keywords

  • Edges
  • Eigenvalues
  • Graph
  • Matrix entries
  • Multiplicity
  • Real symmetric matrix
  • Tree

Mathematics Subject Classification (2010)

  • 15A18
  • 05C50
  • 15B57
  • 13H15
  • 05C05