\input zb-basic
\input zb-ioport
\iteman{io-port 06026831}
\itemau{Bermudo, Sergio; Rodr{\'\i}guez, Jos\'e M.; Sigarreta, Jos\'e M.}
\itemti{Computing the hyperbolicity constant.}
\itemso{Comput. Math. Appl. 62, No. 12, 4592-4595 (2011).}
\itemab
If $X$ is Gromov's $\delta $-hyperbolic space, then the hyperbolicity constant of $X$ is $\inf\{\delta \geq 0: X\text{ is }\delta \text{-hyperbolic}\}$. The authors study graphs with edges of the same length $k$. If $G$ is such graph, then $J(G)$ denotes the union of the set $V(G)$ of vertices of $G$ and the midpoints of the edges of $G$. Let $T_1$ denote the set of all geodesic triangles with vertices in $J(G)$ and which are simple cycles (i.e., simple closed curves). Then $\delta_1(T)=\inf \{\lambda \geq 0:\text{ every triangle }T\in T_1\text{ is }\lambda \text{-thin}\}$. The main result of the paper states that for every graph $G$ with edges of equal length $k$, $\delta_1(G)=\delta (G)$. In addition, it is shown that the hyperbolicity constant $\delta(G)$ is a multiple of $k/4$. As a corollary, the authors derive that for every hyperbolic graph $G$ with edges of length $k$, there is a triangle $T\in T_1$ for which the hyperbolicity constant is attained.
\itemrv{I. G. Nikolaev (Urbana)}
\itemcc{}
\itemut{graph; geodesic metric space; triangle; hyperbolic space; hyperbolicity constant}
\itemli{doi:10.1016/j.camwa.2011.10.041}
\end