arXiv subjects

Vadym Koval

Publications and source records attributed to Vadym Koval.

5 recordsLinked to original sources

arXiv2024

Three observations on the Colin de Verdi\`ere spectral graph parameter

In this small note, we collect several observations pertaining to the famous spectral graph parameter $\mu$ introduced in 1990 by Y. Colin de Verdi\`ere. This parameter is defined as the maximum corank among certain matrices akin to weighted Laplacians; we call them CdV matrices. First, we answer negatively a question mentioned in passing in the influential 1996 survey on $\mu$ by van der Holst, Lov\'asz, and Schrijver concerning the Perron--Frobenious eigenvector of CdV matrices. Second, by definition, CdV matrices posses certain transversality property. In some cases, this property is known to be satisfied automatically. We add one such case to the list. Third, Y. Colin de Verdi\`ere conjectured an upper bound on $\mu(G)$ for graphs embeddable into a fixed closed surface. Following a recent computer-verified counterexample to a continuous version of the conjecture by Fortier Bourque, Gruda-Mediavilla, Petri, and Pineault [arXiv:2312.03504], we also check using computer that the analogous example shows the failure of the conjectured upper bound on $\mu(G)$ for graphs embeddable into 10-torus as well as to several other larger surfaces.

math.CO↗

arXiv2024

A generalization of Clairaut's formula and its applications

The main purpose of this article is to study conditions for a curve on a submanifold $M\subset\mathbb{R}^n$, constructed in a particular way involving the Euclidean distance to $M$, to be a geodesic. We also present the naturally arising generalization of Clairaut's formula needed for the generalization of the main result to higher dimensions.

math.DG↗

arXiv2024

On bundle closures of matrix pencils and matrix polynomials

Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The first main contribution of this paper is to prove that the dimension of each of these bundles is strictly smaller than the dimension of $\mathcal{B}(L)$. The second main contribution is to prove that also the closure of the bundle of a matrix polynomial of grade larger than 1 is the union of the bundle itself with a finite number of other bundles of smaller dimension. To get these results we obtain a formula for the (co)dimension of the bundle of a matrix pencil in terms of the Weyr characteristics of the partial multiplicities of the eigenvalues and of the (left and right) minimal indices, and we provide a characterization for the inclusion relationship between the closures of two bundles of matrix polynomials of the same size and grade.

math.NA↗

arXiv2024

On singular pencils with commuting coefficients

We investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the polynomial of commuting matrices is singular, i.e. its spectrum is the whole complex plane, if and only if (0, 0, ... , 0) belongs to the Taylor spectrum of its coefficients. On the other hand we prove that this equivalence is not longer true if we consider the operators on infinite dimensional Hilbert space as coefficients of polynomial. As a consequence we could propose a new description of (Taylor) spectrum of k-tuple of matrices and we could disprove the conjecture previously proposed in the literature. Additionally, we pointed out the Kronecker forms of the pencils with commuting coefficients.

math.SP↗

arXiv2022

Matrix pencils with the numerical range equal to the whole complex plane

The main purpose of this article is to show that the numerical range of a linear pencil $λA + B$ is equal to $\mathbb{C}$ if and only if $0$ belongs to the convex hull of the joint numerical range of $A$ and $B$. We also prove that if the numerical range of a linear pencil $λA + B$ is equal to $\mathbb{C}$ and $A + A^*, B + B^* \geq 0$, then $A$ and $B$ have a common isotropic vector. Moreover, we improve the classical result which describes Hermitian linear pencils.

math.NA↗