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 eigenvalue of the signless Laplacian of an undirected simple $r$-partite graph on $n$ vertices, $2\le r\le n$. The approach is based on inequalities for the extreme eigenvalues of a block-partitioned Hermitian matrix, established earlier, and on the Rayleigh principle. Specific lower and upper bounds, generalizing and extending known results from $r=2$ to $r\ge2$ are considered, and the cases where these bounds are sharp are described.