Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

Some improvements on the Ky Fan theorem for tensors

The improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

A new estimate for the spectral radius of nonnegative tensors

In this paper, we are concerned with the spectral radius of nonnegative tensors. By estimating the ratio of the smallest component and the largest component of a Perron vector, a new bound for the spectral radius of nonnegative tensors is obtained. It is proved that the new bound improves some existing ones. Finally, a numerical example is implemented to show the effectiveness of the proposed bound.

The spectral radius of maximum weighted cycle

Let [Formula: see text] denote the set of all weighted cycles with vertex set [Formula: see text], edge set [Formula: see text] and positive weight set [Formula: see text]. A weighted cycle [Formula: see text] is called maximum if [Formula: see text] for any [Formula: see text]. In this paper, we give some properties of the Perron vector for the maximum weighted graphs and then determine the maximum weighted cycle in [Formula: see text].

Graph Dynamics of Fuzzy Autocatalytic Set of Fuzzy Graph Type-3 of a Clinical Waste Incineration Process

Fuzzy Autocatalytic Set of Fuzzy Graph Type-3 (FACS) has been successfully implemented in modeling clinical waste incineration process. Six important variables identified in the process are represented as nodes and the catalytic relationships are represented by fuzzy edges in the graph. However, in this paper, graph dynamics of FACS is further investigated using left Perron vector of its transition matrix of fuzzy graph of FACS. This paper will highlight two important variables in the incineration process with regards to the actual process.

A Numerical Algorithm on the Computation of the Stationary Distribution of a Discrete Time Homogenous Finite Markov Chain

The transition matrix, which characterizes a discrete time homogeneous Markov chain, is a stochastic matrix. A stochastic matrix is a special nonnegative matrix with each row summing up to 1. In this paper, we focus on the computation of the stationary distribution of a transition matrix from the viewpoint of the Perron vector of a nonnegative matrix, based on which an algorithm for the stationary distribution is proposed. The algorithm can also be used to compute the Perron root and the corresponding Perron vector of any nonnegative irreducible matrix. Furthermore, a numerical example is given to demonstrate the validity of the algorithm.