let $d$ be a digraph with vertex set $v(d)$. for vertex $vin v(d)$, the degree of $v$,denoted by $d(v)$, is defined as the minimum value if its out-degree and its in-degree.now let $d$ be a digraph with minimum degree $deltage 1$ and edge-connectivity$lambda$. if $alpha$ is real number, then the zeroth-order general randic index is definedby $sum_{xin v(d)}(d(x))^{alpha}$. a digraph is maximall...

Abstract In this paper we consider real or complex skew-Hamiltonian/Hamiltonian pencils λS −H, i.e., pencils where S is a skew-Hamiltonian and H is a Hamiltonian matrix. These pencils occur for example in the theory of continuous time, linear quadratic optimal control problems. We reduce these pencils to canonical and Schur-type forms under structure-preserving transformations, i.e., J-congruen...

In this paper, we further study the GHSS splitting method for nonHermitian positive definite linear problems, which is introduced by Benzi [A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 360-374]. A modified generalization of the Hermitian and skew-Hermitian method (MGHSS) is given, and it still can be used as an effective pr...

Introduced by Okounkov and Reshetikhin in 2003, the Schur Process has been shown to be a determinantal point process, so that each of its correlation functions are determinants of minors of one correlation kernel matrix. In previous papers, this was derived using determinantal expressions of the skew-Schur functions; in this paper, we obtain this result in a different way, using the fact that t...

We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linear-quadratic control problem for partial diierential equations. We develop a structure-preserving skew-Ha...

A new class of the structured matrices related to the discrete skew-self-adjoint Dirac systems is introduced. The corresponding matrix identities and inversion procedure are treated. Analogs of the Schur coefficients and of the Christoffel-Darboux formula are studied. It is shown that the structured matrices from this class are always positive-definite, and applications for an inverse problem f...

An oriented graph G is a simple undirected graph G with an orientation σ, which assigns to each edge of G a direction so that G becomes a directed graph. G is called the underlying graph of G and we denote by S(G) the skew-adjacency matrix of G and its spectrum Sp(G) is called the skew-spectrum of G. In this paper, the skew spectra of two orientations of the Cartesian product of two graphs are ...

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a...