# Graph reduction techniques and the multiplicity of the Laplacian eigenvalues

Graph reduction techniques and the multiplicity of the Laplacian eigenvalues Let M=[mij] be an n×m real matrix, ρ be a nonzero real number, and A be a symmetric real matrix. We denote by D(M) the n×n diagonal matrix diag(∑j=1mm1j,…,∑j=1mmnj) and denote by LAρ the generalized Laplacian matrix D(A)−ρA. A well-known result of Grone et al. states that by connecting one of the end-vertices of P3 to an arbitrary vertex of a graph, does not change the multiplicity of Laplacian eigenvalue 1. We extend this theorem and some other results for a given generalized Laplacian eigenvalue μ. Furthermore, we give two proofs for a conjecture by Saito and Woei on the relation between the multiplicity of some Laplacian eigenvalues and pendant paths. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Linear Algebra and its Applications Elsevier

# Graph reduction techniques and the multiplicity of the Laplacian eigenvalues

, Volume 503 – Aug 15, 2016
18 pages

/lp/elsevier/graph-reduction-techniques-and-the-multiplicity-of-the-laplacian-wHCu3ttEqA
Publisher
Elsevier
ISSN
0024-3795
eISSN
1873-1856
D.O.I.
10.1016/j.laa.2016.04.008
Publisher site
See Article on Publisher Site

### Abstract

Let M=[mij] be an n×m real matrix, ρ be a nonzero real number, and A be a symmetric real matrix. We denote by D(M) the n×n diagonal matrix diag(∑j=1mm1j,…,∑j=1mmnj) and denote by LAρ the generalized Laplacian matrix D(A)−ρA. A well-known result of Grone et al. states that by connecting one of the end-vertices of P3 to an arbitrary vertex of a graph, does not change the multiplicity of Laplacian eigenvalue 1. We extend this theorem and some other results for a given generalized Laplacian eigenvalue μ. Furthermore, we give two proofs for a conjecture by Saito and Woei on the relation between the multiplicity of some Laplacian eigenvalues and pendant paths.

### Journal

Linear Algebra and its ApplicationsElsevier

Published: Aug 15, 2016

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