The average genus for bouquets of circles and dipoles

The bouquet of circles Bn and dipole graph Dn are two important classes graphs in topological theory. For n ≥ 1, we give an explicit formula for the average genus γavg(Bn) Bn. By this expression, one easily sees = (n − ln γ + 1 2) / 2 o(1), where is Euler-Mascheroni constant. Similar results obtained Dn. Our method mainly depends on technique generating series knowledge ordinary differential equations.