The energy EG of a graph G is the sum of the absolute values of the eigenvalues of G. In the case whene G is a molecular graph, EG is closely related to the total ^-electron energy of the corresponding conjugated molecule. We determine the average value of the difference between the energy of two graphs, randomly chosen from the set of all graphs with n vertices and m edges. This result provides a criterion for deciding when two (molecular) graphs are almost coeneigetic.