  We give upper and lower bounds for the spectral radius of a nonnegative
matrix by using its average 2-row sums, and characterize the equality cases if
the matrix is irreducible. We also apply these bounds to various nonnegative
matrices associated with a graph, including the adjacency matrix, the signless
Laplacian matrix, the distance matrix, the distance signless Laplacian matrix,
and the reciprocal distance matrix.

Combinatorics