The average genus for bouquets of circles and dipoles - Zhang, Jinlian (Ars mathematica contemporanea, 2021)
The bouquet of circles ?$B_n$? and dipole graph ?$D_n$? are two important classes of graphs in topological graph theory. For ?$n≥ 1$?, we give an explicit formula for the average genus ?$γ_{avg}(B_n)$? of ?$B_n$?. By this expression, one easily sees ?$γ_{avg}(B_n)=\frac{n-\ln n-γ+1-\ln 2}{2}+o(1)$?, where ?$γ$? is the Euler-Mascheroni constant. Similar results are obtained for ?$D_n$?. Our method is new and deeply depends on the knowledge in ordinary differential equations.
Keywords: average genus; bouquet of circles; dipol; dipole; navadna diferencialna enačba; ordinary differential equation; povprečni rod; šopek krožnic