Normalized Laplacian eigenvalues with chromatic number and independence number of graphs

Let (Formula presented.) be the normalized Laplacian eigenvalues of a graph G with n vertices. Also, let χ and α be the chromatic number and the independence number of a graph G, respectively. In this paper, we discuss some properties of graphs with (Formula presented.). In particular, we characterize all the graphs with (Formula presented.) when the maximum degree is n−1. Moreover, we obtain an upper bound on the multiplicity of normalized Laplacian eigenvalues (Formula presented.) in terms of n and α, and also characterize graphs for which the bound is attained.