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.
Linear Algebra and its Applications – Elsevier
Published: Aug 15, 2016
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