Graph reduction techniques and the multiplicity of the Laplacian eigenvalues

Let M=[mij]M=[mij] be an n×mn×m real matrix, ρ be a nonzero real number, and A   be a symmetric real matrix. We denote by D(M)D(M) the n×nn×n diagonal matrix diag(∑j=1mm1j,…,∑j=1mmnj) and denote by LAρ the generalized Laplacian matrix D(A)−ρAD(A)−ρA. A well-known result of Grone et al. states that by connecting one of the end-vertices of P3P3 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.