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 $δ$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $δ$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$.
If $X$ is hyperbolic, we denote by $δ(X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X) =\inf \{ δ\geq 0:{0.3cm}$ X ${0.2cm}$ $\text{is} {0.2cm} δ\text{-hyperbolic} \}.$ 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 $A(n,m):=\min\{δ(G)\mid G \in \mathcal{G}(n,m) \}$ and $B(n,m):=\max\{δ(G)\mid G \in \mathcal{G}(n,m) \}$. 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.
The quarter-point theorem for Gromov hyperbolic graphs
Florida International University（佛罗里达国际大学）；Facultad de Matemáticas, Universidad Autónoma de Guerrero（格雷罗自治大学数学学院）；Instituto de Física, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学物理研究所）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Information Science, Shandong Technology and Business University（山东工商学院数学与信息科学学院）
