New two-sided bounds for the Perron root and related nonsingularity criteria

New general upper and lower bounds for the Perron root of a nonnegative matrix, which involve nonempty proper subsets of the index set and the matrix sparsity pattern, are suggested, and some special cases are considered. Also the nonsingularity criteria related to the upper bounds presented, which generalize some known results on subclasses of nonsingular $\mathcal{H}$-matrices, are derived.