The Minimal Limit Point of the Third Largest Laplacian Eigenvalues of Graphs

Abstract

Let $G$ be a simple connected graph with $n$ vertices. Denoted by $L(G)$ the Laplacian matrix of G. In this paper, we present a sequence of graphs $G_n$ with $\underset{n\to \mathrm{\infty }}{lim}{\mu }_3(G_n)=1.5550$ by investigating the eigenvalues of the line graphs of $G_n$. Moreover, we prove that the limit is the minimal limit point of the third largest Laplacian eigenvalues of graphs.