# 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

Special Matrices (2017)

• Volume: 5, Issue: 1, page 51-60
• ISSN: 2300-7451

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## Abstract

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We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly Parter vertices. Then, we investigate the change in multiplicity of an eigenvalue based upon a change in an edge value. We show how the multiplicity of the eigenvalue changes depending upon the status of the edge and the edge value. This work explains why, in some cases, edge values have no effect on multiplicities. We also characterize, more precisely, how multiplicity changes with the removal of two adjacent vertices.

## How to cite

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Kenji Toyonaga, and Charles R. Johnson. "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." Special Matrices 5.1 (2017): 51-60. <http://eudml.org/doc/288021>.

@article{KenjiToyonaga2017,
abstract = {We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly Parter vertices. Then, we investigate the change in multiplicity of an eigenvalue based upon a change in an edge value. We show how the multiplicity of the eigenvalue changes depending upon the status of the edge and the edge value. This work explains why, in some cases, edge values have no effect on multiplicities. We also characterize, more precisely, how multiplicity changes with the removal of two adjacent vertices.},
author = {Kenji Toyonaga, Charles R. Johnson},
journal = {Special Matrices},
keywords = {Edges; Eigenvalues; Graph; Matrix entries; Multiplicity; Real symmetric matrix; Tree; edges; eigenvalues; graph; matrix entries; multiplicity; real symmetric matrix; tree},
language = {eng},
number = {1},
pages = {51-60},
title = {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},
url = {http://eudml.org/doc/288021},
volume = {5},
year = {2017},
}

TY - JOUR
AU - Kenji Toyonaga
AU - Charles R. Johnson
TI - 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
JO - Special Matrices
PY - 2017
VL - 5
IS - 1
SP - 51
EP - 60
AB - We take as given a real symmetric matrix A, whose graph is a tree T, and the eigenvalues of A, with their multiplicities. Each edge of T may then be classified in one of four categories, based upon the change in multiplicity of a particular eigenvalue, when the edge is removed (i.e. the corresponding entry of A is replaced by 0).We show a necessary and suficient condition for each possible classification of an edge. A special relationship is observed among 2-Parter edges, Parter edges and singly Parter vertices. Then, we investigate the change in multiplicity of an eigenvalue based upon a change in an edge value. We show how the multiplicity of the eigenvalue changes depending upon the status of the edge and the edge value. This work explains why, in some cases, edge values have no effect on multiplicities. We also characterize, more precisely, how multiplicity changes with the removal of two adjacent vertices.
LA - eng
KW - Edges; Eigenvalues; Graph; Matrix entries; Multiplicity; Real symmetric matrix; Tree; edges; eigenvalues; graph; matrix entries; multiplicity; real symmetric matrix; tree
UR - http://eudml.org/doc/288021
ER -

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