Bounds on Gromov Hyperbolicity Constant

TL;DR

This paper investigates bounds on the Gromov hyperbolicity constant for graphs, providing exact values for minimal bounds, estimates for maximal bounds, and applications to random graphs, advancing understanding of hyperbolic properties in graph theory.

Contribution

The paper derives bounds for hyperbolicity constants in graphs, computes exact minimal values for all parameters, and applies findings to random graph models, offering new insights into graph hyperbolicity.

Findings

Exact values of minimal hyperbolicity constants for all graph parameters.

Good bounds for maximal hyperbolicity constants in graphs.

Application of bounds to analyze random graphs.

Abstract

If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} \in X$, a geodesic triangle $T=\{x_{1},x_{2},x_{3}\}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $\delta$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\delta(X) =\inf \{ \delta\geq 0:{0.3cm}$ X ${0.2cm}$ $\text{is} {0.2cm} \delta \text{-hyperbolic} \}.$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by $\mathcal{G}(n,m)$ the set of graphs $G$ with $n$ vertices and $m$ edges, and such that every edge has length $1$. In this work we estimate $A(n,m):=\min\{\delta(G)\mid G \in \mathcal{G}(n,m) \}$ and $B(n,m):=\max\{\delta(G)\mid G \in \mathcal{G}(n,m) \}$. In particular, we obtain good bounds for $B(n,m)$, and we compute the precise value of $A(n,m)$ for all values of $n$ and $m$. Besides, we apply these results to random graphs.