  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.
