On the Spectrum of Laplacian Matrix

Abstract

Let $G$ be a simple graph of order $n$ . The matrix $\mathrm{ℒ}\left(G\right)=D\left(G\right)-A\left(G\right)$ is called the Laplacian matrix of $G$ , where $D\left(G\right)$ and $A\left(G\right)$ denote the diagonal matrix of vertex degrees and the adjacency matrix of $G$ , respectively. Let $l_1\left(G\right)$ , $l_{n-1}\left(G\right)$ be the largest eigenvalue, the second smallest eigenvalue of $\mathrm{ℒ}\left(G\right)$ respectively, and ${\lambda }_1\left(G\right)$ be the largest eigenvalue of $A\left(G\right)$ . In this paper, we will present sharp upper and lower bounds for $l_1\left(G\right)$ and $l_{n-1}\left(G\right)$ . Moreover, we investigate the relation between $l_1\left(G\right)$ and ${\lambda }_1\left(G\right)$ .