Proof of a conjecture on `plateaux' phenomenon of graph Laplacian eigenvalues

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 $\ge 3$, 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 $\ge 3$ 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.