For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(AD) ≥ r(AD) and r(A) r(AD) ≥ r(AD), leave open the question of what determines the order of r(AD) with respect to r(A) r(AD). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that dete...

We consider a multidimensional optimization problem in the framework of tropical mathematics. The problem is formulated to minimize a nonlinear objective function that is defined on vectors in a finitedimensional semimodule over an idempotent semifield and calculated by means of multiplicative conjugate transposition. We offer two complete direct solutions to the problem. The first solution con...

For decades considerable efforts have been exerted to resolve the inverse eigenvalue problem for nonnegative matrices. Yet fundamental issues such as the theory of existence and the practice of computation remain open. Recently it has been proved that, given an arbitrary (n − 1)-tuple L = (λ2, . . . , λn) ∈ C n−1 whose components are closed under complex conjugation, there exists a unique posit...

For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(AD) ≥ r(AD) and r(A) r(AD) ≥ r(AD), leave open the question of what determines the order of r(AD) with respect to r(A) r(AD). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that dete...

This paper discusses about the solution of fuzzy Volterra integral equation of first-kind (F-VIE1) using spectral method. The parametric form of fuzzy driving term is applied for F-VIE1, then three classifications for (F-VIE1) are searched to solve them. These classifications are considered based on the interval sign of the kernel. The Gauss-Legendre points and Legendre weights for arithmetics ...

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper we present the spectral analogues of Erdős’ theorem and Moon-Moser’s theorem, respectively. Let Gk n be the ...

In this paper we use preferred and quasi-preferred bases of generalized eigenspaces associated with the spectral radius of nonnegative matrices to analyze the existence and uniqueness of a variant of the Jordan canonical form which we call FrobeniusJordan form. It is a combination of the classical Jordan canonical form in the part associated with the eigenvalues that are different from the spec...

The spectral radius of the adjacency matrix can impact both algorithmic efficiency as well as the stability of solutions to an underlying dynamical process. Although much research has considered the distribution of the spectral radius for undirected random graph models, as symmetric adjacency matrices are amenable to spectral analysis, very little work has focused on directed graphs. Consequent...

In this paper, we show the evaluation of the spectral radius for node degree as the basis to analyze the variation in the node degrees during the evolution of scale-free networks and small-world networks. Spectral radius is the principal eigenvalue of the adjacency matrix of a network graph and spectral radius ratio for node degree is the ratio of the spectral radius and the average node degree...