Abstract
Abstract In this paper, some estimations for the spectral radius of nonnegative matrices and the smallest eigenvalue of M-matrices are given by matrix directed graphs and their k-path covering. The existent results on the upper and lower bounds of the spectral radius of nonnegative matrices are improved. MSC:15A18, 65F15.
Highlights
The following notations are used throughout this paper
A[P] denotes the prime sub-matrices of A where row-column subscripts are all in P ⊆ N, ρ(A) is the spectral radius of A, and ω (A) is the smallest eigenvalue according to the module of A
With regard to estimations for the nonnegative matrix spectral radius, the earliest result is given by Perron-Frobenius, that is, min i∈N
Summary
The following notations are used throughout this paper. Let A = (aij) ∈ Rn×n be an n × n matrix with real entries. We denote it A ∈ Nn. If A = sI – B, B ∈ Nn, s > ρ(B), A is called a nonsingular M-matrix. With regard to estimations for the nonnegative matrix spectral radius, the earliest result is given by Perron-Frobenius (see [ ]), that is, min i∈N Let A be a nonsingular M-matrix and denote s = maxi∈N {aii}.
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