Bounds for the extreme eigenvalues of the laplacian and signless laplacian of a graph

The paper suggests a new approach to deriving lower bounds for the Laplacian spectral radius and upper bounds for the smallest eigenvalve of the signless Laplacian of an undirected simple r-partite graph on n vertices, 2 ≤ r ≤ n. The approach is based on inequalities for the extreme eigenvalves of a block-partitioned Hermitian matrix, established earlier, and on the Rayleigh principle. Specific lower and upper bounds generalizing known results and extending them from r = 2 to r ≥ 2 are considered, and the cases where these bounds are sharp are described. Bibliography: 13 titles. © 2012 Springer Science+Business Media, Inc.