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 $\beta$ and the length of $\beta$ in the Cayley graph of a finitely presented group $G$ is bounded above then $G$ is hyperbolic. We plan to use these results to characterize hyperbolic groups in terms of random walks.