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

TL;DR

This paper proves a conjecture relating the multiplicity of Laplacian eigenvalues to the structure of pendant paths in graphs, advancing understanding of eigenvalue multiplicities in graph theory.

Contribution

The paper establishes a general theorem for Laplacian and signless Laplacian eigenvalues that confirms a conjecture about eigenvalue multiplicities based on pendant path structures.

Findings

Proved a conjecture linking eigenvalue multiplicity to pendant path counts.

Established a more general theorem applicable to Laplacian and signless Laplacian matrices.

Enhanced understanding of spectral properties related to graph structures.

Abstract

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.