Sharp bounds for the spectral radius of nonnegative matrices

We give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with all row sums positive using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We compare these bounds with the known bounds using the row sums by examples. We also apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix and some distance-based matrices. Some known results are generalized and improved.