Some new bounds on the spectral radius of nonnegative matrices

Sharp Bounds on the Spectral Radius of Nonnegative Matrices and Comparison to the Frobenius’ Bounds

“…In case of A is symmetric ( ) ρ A is the largest eigenvalue of A . For 1 ≤ ≤ i n, the i-th row sum of A , denoted by ( ) (1) In recent decades, the problem of bounding the largest eigenvalue in modulus of a nonnegative matrix has attracted the interest of many researchers (see, [1-4, 7-8, 14-20] and the references therein), since Perron-Frobenius theory plays an important role in the mathematical fields of dynamical systems, cryptography, control and graph theory (see, e.g. [1-2, 4, 7-8, 12-20]).…”

“…Algebraic methods for the computation of the bounds of the spectral radius of a nonnegative matrix have been presented in [1,4,8,[14][15]18] giving applications in the spectral radii of the various matrices of a graph or a digraph, including the signless Laplacian spectral radius, the distance spectral radius and the distance signless Laplacian spectral radius, the spectral radius of the reciprocal distance matrix (see, [1,8,[14][15]18] and the references therein). The bounds of the spectral radius, which provided in the aforementioned studies, depend on the size, the elements, the row sum and various average row sums of the nonnegative matrix and their computation is more complicated than the computation of the largest eigenvalue of the matrix.…”