On the second real eigenvalue of nonnegative and Z-matrices

We give bounds for the second real eigenvalue of nonegative matrices and Z-matrices. Furthermore, we establish upper bounds for the maximal spectral radii of principal submatrices of nonnegative matrices. Using these bounds, we prove that our inequality for the second real eigenvalue of the adjacency matrix of a connected regular graph improves a well-known bound for the second eigenvalue using Cheeger's inequality. (C) Elsevier Science Inc., 1997.