Graph reduction techniques and the multiplicity of the Laplacian eigenvalues

Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $\rho$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by $L_{A}^{\rho}$ the generalized Laplacian matrix $D(A)-\rho A$. A well-known result of Grone et al. states that by connecting one of the end-vertices of $P_{3}$ 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 $\mu$. 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.