An Inequality for the Perron Pair of an Irreducible and Symmetric Nonnegative Tensor with Application

The main purpose of this paper is to consider the Perron pair of an irreducible and symmetric nonnegative tensor and the smallest eigenvalue of an irreducible and symmetric nonsingular M-tensor. We analyze the analytical property of an algebraic simple eigenvalue of symmetric tensors. We also derive an inequality about the Perron pair of nonnegative tensors based on plane stochastic tensors. We finally consider the perturbation of the smallest eigenvalue of nonsingular M-tensors and design a strategy to compute its smallest eigenvalue. We verify our results via random numerical examples.