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BY 4.0 license Open Access Published by De Gruyter Open Access December 13, 2019

The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

  • Kenji Toyonaga EMAIL logo
From the journal Special Matrices

Abstract

Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

MSC 2010: 15A18; 05C50; 13H15; 05C05

References

[1] R. Horn, C.R.Johnson, Matrix Analysis, 2nd Edition. Cambridge University Press, New York, 2013.Search in Google Scholar

[2] C.R. Johnson, A. Leal-Duarte and C.M. Saiago, The Parter-Wiener theorem: refinement and generalization, SIAM J. Matrix Anal. Appl. 25 (2003), 352-361.10.1137/S0895479801393320Search in Google Scholar

[3] C.R. Johnson, A. Leal-Duarte, C.M. Saiago. The change in eigenvalue multiplicity associated with perturbation of a diagonal entry, Linear Multilinear Algebra 60 (2012), 525-532 .10.1080/03081087.2011.610973Search in Google Scholar

[4] C.R. Johnson, P.R. McMichael, The change in multiplicity of an eigenvalue of a Hermitian matrix associated with the removal of an edge from its graph, Discrete Math. 311 (2011), 166-170.10.1016/j.disc.2010.10.010Search in Google Scholar

[5] C.R. Johnson, C. M. Saiago, Geometric Parter-Wiener, etc. theory, Linear Algebra Appl, 537 (2018), 332-347.10.1016/j.laa.2017.09.035Search in Google Scholar

[6] C.R. Johnson, C.M. Saiago, Eigenvalues, Multiplicities and Graphs. Cambridge University Press, New York, 2018.10.1017/9781316155158Search in Google Scholar

[7] K. Toyonaga, C.R. Johnson, The classification of edges and the change in multiplicity of an eigenvalue of a real symmetric matrix tesulting from the change in an edge value, Spec. Matrices 5 (2017), 51-60.10.1515/spma-2017-0004Search in Google Scholar

[8] C.R. Johnson, C.M. Saiago, K. Toyonaga, Classification of vertices and edges with respect to the geometric multiplicity of an eigenvalue in a matrix, with a given graph, over a field, Linear Multilinear Algebra 66 (2018), 2168-2182.10.1080/03081087.2017.1389848Search in Google Scholar

Received: 2019-09-30
Accepted: 2019-11-29
Published Online: 2019-12-13

© 2019 Kenji Toyonaga, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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