Counting connected graphs with large excess

proceedings of the 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.