Related papers: Distance in the curve graph

We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura…

With a view on graph clustering, we present a definition of vertex-to-vertex distance which is based on shared connectivity. We argue that vertices sharing more connections are closer to each other than vertices sharing fewer connections.…

We derive integral and sup-estimates for the curvature of stably marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature…

Comparison of graph structure is a ubiquitous task in data analysis and machine learning, with diverse applications in fields such as neuroscience, cyber security, social network analysis, and bioinformatics, among others. Discovery and…

Consider a finite directed graph without cycles in which the arrows are weighted. We present an algorithm for the computation of a new distance, called path-length-weighted distance, which has proven useful for graph analysis in the context…

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices at…

We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound in terms of the degree the number of topological types of singular algebraic curves in the real projective plane.

This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces.

Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…

Let $\mathbb{F}_q$ denote a finite field with $q$ elements. Let $n,k$ denote integers with $n>2k\geq 6$. Let $V$ denote a vector space over $\mathbb{F}_{q}$ that has dimension $n$. The vertex set of the Grassmann graph $J_q(n,k)$ consists…

It is proved the generalization of Toponogov theorem about the length of the curve in two-dimensional Riemannian manifolds in the case of two-dimensional Alexandrov spaces.

A new tur\'an-type problem on distances on graphs was introduced by Tyomkyn and Uzzell. In this paper, we focus on the case that the distance is two. We primely show that for any value of $n$, a graph on $n$ vertices without three vertices…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

If we are given a connected finite graph $G$ and a subset of its vertices $V_{0}$, we define a distance-residual graph as a graph induced on the set of vertices that have the maximal distance from $V_{0}$. Some properties and examples of…

We report on some findings concerning Connes' noncommutative distance $d$ on a weighted undirected graph $G$. Our main result is the lower bound $\ell/\Delta(G)\le d$ where $\ell$ is the geodesic distance and $\Delta(G)$ the degree of $G$.…

We show there exists a closed graph manifold $N$ and infinitely many non-separable, horizontal surfaces $\{S_{n} \to N\}_{n \in \mathbb{N}}$ such that there does not exist a quasi-isometry $\pi_1(N) \to \pi_1(N)$ taking $\pi_1(S_{n})$ to…

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…

Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal…

A graph is called primitive if its automorphism group acts primitively on the vertex set. In this paper, we prove a classification of the possible distance sequences of locally infinite primitive graphs. In particular we show that if a…

We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not…