Related papers: Intersection numbers in the curve graph with a uni…
We study the relationship between the lengths of closed geodesics on hyperbolic surfaces and their topological complexity, measured by the self-intersection number. In particular, we provide explicit upper bounds for the length $s_k(X)$ of…
We survey theoretical, algorithmic, and computational results at the intersection of distance geometry problems and mathematical programming, both with and without adjacencies as part of the input. While mathematical programming methods can…
We describe a polynomial-time algorithm to compute a (tight) geodesic between two curves in the curve graph. As well as enabling us to compute the distance between a pair of curves, this has several applications to mapping classes. For…
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are…
We formulate a uniqueness conjecture for curve shortening flow of proper curves on certain symmetric surfaces and give an example of a non-flat metric on the plane with respect to which curve shortening flow is not unique. That is, with…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging.…
We present a simplified exposition of some classical and modern results on graph drawings in the plane. These results are chosen so that they illustrate some spectacular recent higher-dimensional results on the border of topology and…
A mixed graph can be seen as a type of digraph containing some edges (two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures…
The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among…
In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…
We consider a dynamical system on a metric graph, that corresponds to a semiclassical solution of a time-dependent Schr\"odinger equation. We omit all details concerning mathematical physics and work with a purely discrete problem. We find…
Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time…
We bound the genus of a projective curve lying on a complete intersection surface in terms of its degree and the degrees of the defining equations of the surface on which it lies.
We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-P\'olya-Weinberger, and Yang types, among others. We show that the sharp constants of…
Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…
We show that if a graph contains few induced copies of a given graph then its edges are distribited unevenly.
For two oriented simple closed curves on a compact orientable surface with a connected boundary we introduce a simple computation of a value in the first homology group of the surface, which detects in some cases that the geometric…
We give explicit bounds on the intersection number between any curve on a tight multigeodesic and the two ending curves. We use this to construct all tight multigeodesics and so conclude that distances in the curve graph are computable. The…
We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these…