Summary: 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 \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) 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 \(X\).
We denote by \(\delta(X)\) the sharp hyperbolicity constant of \(X\), i.e. \(\delta(X):= \text{inf}\{\delta\geq 0 : X\) is \(\delta\)-hyperbolic}. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.
The main aim of this paper is to obtain information about the hyperbolicity constant of the complement graph \(\overline G\) in terms of properties of the graph \(G\). In particular, we prove that if \(\text{diam}(V(G))\geq 3\), then \(\delta(\overline G)\leq 2\), and that the inequality is sharp. Furthermore, we find some Nordhaus-Gaddum type results on the hyperbolicity constant of a graph \(\delta(G)\).