Integral criteria of hyperbolicity for graphs and groups

We establish three criteria of hyperbolicity of a graph in terms of “average width of geodesic bigons”. In particular we prove that if the ratio of the Van Kampen area of a geodesic bigon \(\beta \) $\beta$ and the length of \(\beta \) $\beta$ in the Cayley graph of a finitely presented group G is bounded above then G is hyperbolic.