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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

Article

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.

Keywords

Edges Eigenvalues Graph Matrix entries Multiplicity Real symmetric matrix Tree 

Mathematics Subject Classification (2010)

15A18 05C50 15B57 13H15 05C05 

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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

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