  Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of
its end vertices has degree $1$, the other end has degree $\ge3$, and all the
internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths
of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$
which are an end vertex of some pendant paths of length $k$. Motivated by the
problem of characterizing dendritic trees, N. Saito and E. Woei conjectured
that any graph $G$ has some Laplacian eigenvalue with multiplicity at least
$p_k(G)-q_k(G)$. We prove a more general result for both Laplacian and signless
Laplacian eigenvalues from which the conjecture follows.
