This paper is concerned with bounds of eigenvalues of a complex matrix. Both lower and upper bounds of modulus of eigenvalues are given by the Stein equation. Furthermore, two sequences are presented which converge to the minimal and the maximal modulus of eigenvalues, respectively. We have to point out that the two sequences are not recommendable for practical use for finding the minimal and the maximal modulus of eigenvalues.
We are concerned with bounds of the matrix eigenvalues and its exponential. Combining the Lyapunov equation with the weighted logarithmic matrix norm technique, four sequences are presented to locate eigenvalues of a matrix. Based on the relations between the real parts of the eigenvalues and the weighted logarithmic matrix norms, we derive both lower and upper bounds of the matrix exponential, which complement and improve the existing results in the literature. Some numerical examples are also...
A total dominating set in a graph is a subset of such that each vertex of is adjacent to at least one vertex of . The total domination number of is the minimum cardinality of a total dominating set. A function is a signed dominating function (SDF) if the sum of its function values over any closed neighborhood is at least one. The weight of an SDF is the sum of its function values over all vertices. The signed domination number of is the minimum weight of an SDF on . In this paper...
Let be an symmetric, irreducible, and nonnegative matrix whose eigenvalues are . In this paper we derive several lower and upper bounds, in particular on and , but also, indirectly, on . The bounds are in terms of the diagonal entries of the group generalized inverse, , of the singular and irreducible M-matrix . Our starting point is a spectral resolution for . We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected...
Felsenstein's classical model for Gaussian distributions on a phylogenetic tree is shown to be a toric variety in the space of concentration matrices. We present an exact semialgebraic characterization of this model, and we demonstrate how the toric structure leads to exact methods for maximum likelihood estimation. Our results also give new insights into the geometry of ultrametric matrices.