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 \(\lambda_i(L)\), \(i = 1, 2, \ldots, n\), 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 \(n\), \(lambda_2(L) = \lambda_3(L) = \cdots = \lambda_{n-1}(L)\) 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.