On the Normalized Laplacian Eigenvalues of Graphs

Abstract

Let $G=(V,E)$ be a simple connected graph with $n$ vertices and $m$ edges. Further, let ${\lambda }_i(L)$, $i=1,2,\dots ,n$, be the non-increasing eigenvalues of the normalized Laplacian matrix of the graph $G$. In this paper, we obtain the following result: For a connected graph $G$ of order $n$, $lambd{a}_2(L)={\lambda }_3(L)=\cdots ={\lambda }_{n-1}(L)$ if and only if $G$ is a complete graph $K_n$ or $G$ is a complete bipartite graph $K_{p,q}$. Moreover, we present lower and upper bounds for the normalized Laplacian spectral radius of a graph and characterize graphs for which the lower or upper bounds are attained.