arXiv subjects

Alexandr Polyanskii

Publications and source records attributed to Alexandr Polyanskii.

At least 19 recordsLinked to original sources

arXiv2023

Alon-Boppana-type bounds for weighted graphs

The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph. The weighted degree of a vertex in a weighted graph is the sum of weights of edges incident to the vertex. A weighted graph is called regular if the weighted degrees of its vertices are the same. Using the result on unraveled balls, we prove a variation of the Alon-Boppana theorem for regular weighted graphs.

math.CO↗

arXiv2023

Intersecting diametral balls induced by a geometric graph II

For a graph whose vertices are points in $\mathbb R^d$, consider the closed balls with diameters induced by its edges. The graph is called a Tverberg graph if these closed balls intersect. A max-sum tree of a finite point set $X \subset \mathbb R^d$ is a tree with vertex set $X$ that maximizes the sum of Euclidean distances of its edges among all trees with vertex set $X$. Similarly, a max-sum matching of an even set $X \subset \mathbb R^d$ is a perfect matching of $X$ maximizing the sum of Euclidean distances between the matched points among all perfect matchings of $X$. We prove that a max-sum tree of any finite point set in $\mathbb R^d$ is a Tverberg graph, which generalizes a recent result of Abu-Affash et al., who established this claim in the plane. Additionally, we provide a new proof of a theorem by Bereg et al., which states that a max-sum matching of any even point set in the plane is a Tverberg graph. Moreover, we proved a slightly stronger version of this theorem.

math.CO↗

arXiv2022

Intersecting ellipses induced by a max-sum matching

For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's endpoints and eccentricity $\sqrt 3 / 2$. Using an optimization approach, we prove that the convex sets bounded by these ellipses intersect, answering a Tverberg-type question of Andy Fingerhut from 1995.

cs.CG↗

arXiv2022

Extensions of polynomial plank covering theorems

We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths.

math.MG↗

arXiv2021

Covering by planks and avoiding zeros of polynomials

We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball we show that for every real nonzero $d$-variate polynomial $P$ of degree $n$, there exists a point in the unit $d$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $P$. Using the polynomial approach, we also prove the strengthening of the Fejes T\'oth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $\pi$.

math.MG↗

arXiv2021

Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-\lambda$ can be defined by a finite set of forbidden induced subgraphs if and only if $\lambda < \lambda^*$, where $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, and $\rho$ is the unique real root of $x^3 = x + 1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman's work on those limit points in $[-2, \infty)$. We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by $N_{\alpha, \beta}(n)$ the maximum number of unit vectors in $\mathbb{R}^d$ where all pairwise inner products lie in $\{\alpha, \beta\}$ with $-1 \le \beta < 0 \le \alpha < 1$. Very recently Jiang, Tidor, Yao, Zhang and Zhao determined the limit of $N_{\alpha, \beta}(d)/d$ as $d\to\infty$ when $\alpha + 2\beta < 0$ or $(1-\alpha)/(\alpha-\beta) \in \{1,\sqrt2,\sqrt3\}$, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever $(1-\alpha)/(\alpha - \beta) < \lambda^*$.

math.CO↗

arXiv2021

Disjoint edges in geometric graphs

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the convex hull of its neighbours. We show that for a geometric graph with $n$ vertices and $e$ edges there are at least $\frac{n}{2}\binom{2e/n}{3}$ pairs of disjoint edges provided that $2e\geq n$ and all the vertices of the graph are pointed. Besides, we prove that if any edge of a geometric graph with $n$ vertices is disjoint from at most $ m $ edges, then the number of edges of this graph does not exceed $n(\sqrt{1+8m}+3)/4$ provided that $n$ is sufficiently large. These two results are tight for an infinite family of graphs.

math.CO↗

arXiv2021

Intersecting diametral balls induced by a geometric graph

For a graph whose vertex set is a finite set of points in the Euclidean $d$-space consider the closed (open) balls with diameters induced by its edges. The graph is called a (an open) Tverberg graph if these closed (open) balls intersect. Using the idea of halving lines, we show that ($i$) for any finite set of points in the plane, there exists a Hamiltonian cycle that is a Tverberg graph; ($ii$) for any $ n $ red and $ n $ blue points in the plane, there exists a perfect red-blue matching that is a Tverberg graph. Also, we prove that ($iii$) for any even set of points in the Euclidean $ d $-space, there exists a perfect matching that is an open Tverberg graph; ($iv$) for any $ n $ red and $ n $ blue points in the Euclidean $ d $-space, there exists a perfect red-blue matching that is a Tverberg graph.

math.CO↗

arXiv2020

The extremal number of surfaces

In 1973, Brown, Erd\H{o}s and S\'os proved that if $\mathcal{H}$ is a 3-uniform hypergraph on $n$ vertices which contains no triangulation of the sphere, then $\mathcal{H}$ has at most $O(n^{5/2})$ edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan, and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface $\mathcal{S}$.

math.CO↗

arXiv2020

A cap covering theorem

A cap of spherical radius $\alpha$ on a unit $d$-sphere $S$ is the set of points within spherical distance $\alpha$ from a given point on the sphere. Let $\mathcal F$ be a finite set of caps lying on $S$. We prove that if no hyperplane through the center of $ S $ divides $\mathcal F$ into two non-empty subsets without intersecting any cap in $\mathcal F$, then there is a cap of radius equal to the sum of radii of all caps in $\mathcal F$ covering all caps of $\mathcal F$ provided that the sum of radii is less $\pi/2$. This is the spherical analog of the so-called Circle Covering Theorem by Goodman and Goodman and the strengthening of Fejes T\'oth's zone conjecture proved by Jiang and the author arXiv:1703.10550.

math.MG↗

arXiv2019

Approximation of the average of some random matrices

Rudelson's theorem states that if for a set of unit vectors $u_i$ and positive weights $c_i$, we have that $\sum c_i u_i\otimes u_i$ is the identity operator $I$ on ${\mathbb R}^d$, then the sum of a random sample of $Cd\ln d$ of these diadic products is close to $I$. The $\ln d$ term cannot be removed. On the other hand, the recent fundamental result of Batson, Spielman and Srivastava and its improvement by Marcus, Spielman and Srivastava show that the $\ln d$ term can be removed, if one wants to show the existence of a good approximation of $I$ as the average of a few diadic products. It is known that essentially the same proof as Rudelson's yields a more general statement about the average of positive semi-definite matrices. First, we give an example of an average of positive semi-definite matrices where there is no approximation of this average by $Cd$ elements. Thus, the result of Batson, Spielman and Srivastava cannot be extended to this wider class of matrices. Next, we present a stability version of Rudelson's result on positive semi-definite matrices, and thus, extend it to certain non-symmetric matrices. This yields applications to the study of the Banach--Mazur distance of convex bodies. Finally, we show that in some cases, one needs to take a subset of the vectors of order $d^2$ to approximate the identity.

math.FA↗

arXiv2019

Perron and Frobenius meet Carath\'{e}odory

We present a new approach of proving certain Carath\'{e}odory-type theorems using the Perron-Frobenius Theorem, a classical result in matrix theory describing the largest eigenvalue of a matrix with positive entries. One of the problems left open in this note is whether our approach may be extended to prove similar results in the area, in particular the Colourful Carath\'{e}odory Theorem.

math.MG↗

arXiv2017

On almost-equidistant sets - II

A set in $\mathbb R^d$ is called almost-equidistant if for any three distinct points in the set, some two are at unit distance apart. First, we give a short proof of the result of Bezdek and L\'angi claiming that an almost-equidistant set lying on a $(d-1)$-dimensional sphere of radius $r$, where $r<1/\sqrt{2}$, has at most $2d+2$ points. Second, we prove that an almost-equidistant set $V$ in $\mathbb R^d$ has $O(d)$ points in two cases: if the diameter of $V$ is at most $1$ or if $V$ is a subset of a $d$-dimensional ball of radius at most $1/\sqrt{2}+cd^{-2/3}$, where $c<1/2$. Also, we present a new proof of the result of Kupavskii, Mustafa and Swanepoel arXiv:1708.01590 that an almost-equidistant set in $\mathbb R^d$ has $O(d^{4/3})$ elements.

math.MG↗

arXiv2017

Forbidden subgraphs for graphs of bounded spectral radius, with applications to equiangular lines

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let $\mathcal{F}(\lambda)$ be the family of connected graphs of spectral radius $\le \lambda$. We show that $\mathcal{F}(\lambda)$ can be defined by a finite set of forbidden subgraphs if and only if $\lambda < \lambda^* := \sqrt{2+\sqrt{5}} \approx 2.058$ and $\lambda \not\in \{\alpha_2, \alpha_3, \dots\}$, where $\alpha_m = \beta_m^{1/2} + \beta_m^{-1/2}$ and $\beta_m$ is the largest root of $x^{m+1}=1+x+\dots+x^{m-1}$. The study of forbidden subgraphs characterization for $\mathcal{F}(\lambda)$ is motivated by the problem of estimating the maximum cardinality of equiangular lines in the $n$-dimensional Euclidean space $\mathbb{R}^n$ --- a family of lines through the origin such that the angle between any pair of them is the same. Denote by $N_\alpha(n)$ the maximum number of equiangular lines in $\mathbb{R}^n$ with angle $\arccos\alpha$. We establish the asymptotic formula $N_\alpha(n) = c_\alpha n + O_\alpha(1)$ for every $\alpha \ge \frac{1}{1+2\lambda^*}$. In particular, $N_{1/3}(n) = 2n+O(1)$ and $N_{1/5}(n), N_{1/(1+2\sqrt{2})}(n) = \frac{3}{2}n+O(1)$. Besides we show that $N_\alpha(n) \le 1.49n + O_\alpha(1)$ for every $\alpha \neq \tfrac{1}{3}, \tfrac{1}{5}, \tfrac{1}{1+2\sqrt{2}}$, which improves a recent result of Balla, Dr\"axler, Keevash and Sudakov.

math.CO↗

arXiv2017

On almost-equidistant sets

A finite set of points in $\mathbb R^d$ is called almost-equidistant if among any three distinct points in the set, some two are at unit distance. We prove that an almost-equidistant set in $\mathbb R^d$ has cardinality at most $5d^{13/9}$.

math.MG↗

arXiv2017

Proof of L\'aszl\'o Fejes T\'oth's zone conjecture

A zone of width $\omega$ on the unit sphere is the set of points within spherical distance $\omega/2$ of a given great circle. We show that the total width of any collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T\'oth from 1973.

math.MG↗

arXiv2017

Hunting for reduced polytopes

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove a novel necessary condition on reduced polytopes in three-dimensional Euclidean space.

math.MG↗

arXiv2016

Helly-type theorem for eigenvectors

We prove that if any $\lfloor3d/2 \rfloor$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ ($\mathbb K$ is an arbitrary field) have a common eigenvector then all operators in the family have a common eigenvector. Moreover, $\lfloor 3d/2\rfloor$ cannot be replaced by a smaller number. Also, we study the following problem, achieving partial results: prove that if any $l=O(d)$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ have a common non-trivial invariant subspace then all operators in the family have a common non-trivial invariant subspace.

math.MG↗