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
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.
KeywordsEdges Eigenvalues Graph Matrix entries Multiplicity Real symmetric matrix Tree
Mathematics Subject Classification (2010)15A18 05C50 15B57 13H15 05C05
- 1.Horn, R., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)Google Scholar