Related papers: On the hyperbolicity constant of circular-arc grap…

This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a…

A graph is called $\alpha_i$-metric ($i \in {\cal N}$) if it satisfies the following $\alpha_i$-metric property for every vertices $u, w, v$ and $x$: if a shortest path between $u$ and $w$ and a shortest path between $x$ and $v$ share a…

In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying…

In this work we study the asymptotic traffic flow in Gromov's hyperbolic graphs. We prove that under certain mild hypotheses the traffic flow in a hyperbolic graph tends to pass through a finite set of highly congested nodes. These nodes…

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity,…

Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a "worst-case" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation…

We show that a geodesic metric space is hyperbolic in the sense of Gromov if and only if intersections of balls have bounded eccentricity. In particular, $\R$-trees are characterized among geodesic metric spaces by the property that the…

We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that…

The concept of Gromov hyperbolicity manifests itself in many different ways. With only mild assumptions on the underlying metric space, the spectrum of equivalent properties includes various thin triangle conditions, the stability of…

We show that expander graphs must have Gromov-hyperbolicity at least proportional to their diameter, with a constant of proportionality depending only on the expansion constant and maximal degree. In other words, expanders contain geodesic…

We study bond percolation for a family of infinite hyperbolic graphs. We relate percolation to the appearance of homology in finite versions of these graphs. As a consequence, we derive an upper bound on the critical probabilities of the…

We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of $F_N$ for free products of groups, and show their hyperbolicity. Given a countable group $G$ which splits as…

Given a compact surface $\Sigma$ with boundary and a relation $\Gamma$ on $\pi_0(\partial\Sigma)$, we define the prescribed arc graph $\mathscr A(\Sigma,\Gamma)$ to be the full subgraph of the arc graph $\mathscr A(\Sigma)$ containing only…

We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly…

We introduce a notion of a girth-regular graph as a $k$-regular graph for which there exists a non-descending sequence $(a_1, a_2, \dots, a_k)$ (called the signature) giving, for every vertex $u$ of the graph, the number of girth cycles the…

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichm\"uller space, Hitchin representations and geodesic currents. We add to…

In this article, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous article, we determine precisely when a graph braid group is Gromov-hyperbolic,…

Given a surface, the fine $k$-curve graph of the surface is a graph whose vertices are simple closed essential curves and whose edges connect curves that intersect in at most $k$ points. We note that the fine $k$-curve graph is hyperbolic…

Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ…

Hyperbolicity is a graph parameter related to how much a graph resembles a tree with respect to distances. Its computation is challenging as the main approaches consist in scanning all quadruples of the graph or using fast matrix…