Normalized Laplacian spectrum of complete multipartite graphs

The spectrum of the normalized Laplacian matrix of a graph provides many structural information of the graph, and it has many applications in numerous areas and in different guises. Let G be a complete k-partite graph with k≥3. In this paper, we give the necessary and sufficient condition for G which is determined by their normalized Laplacian spectrum. Moreover, we obtain a majorization theory of normalized Laplacian spectral radius of G, which enables us to find the maximal and minimal normalized Laplacian spectral radii among all complete k-partite graphs with fixed order, respectively.