Labelled bicyclic graph count ($C(n,n+1)=\Theta (n^{n+1})(n\ge 4)$)

If $b_k$ counts labelled bicyclic cores on $k$ vertices, the bicyclic kernel classification gives $b_k\le Ak!k^2$. Three internally nonempty paths between two branching vertices give $b_k\ge k!\left(\genfrac{}{}{0ex}{}{k-3}{2}\right)/12$ for $k\ge 5$. Attach a forest with those core vertices as roots to obtain $C(n,n+1)={\sum }_k\left(\genfrac{}{}{0ex}{}{n}{k}\right)b_{k}k{n}^{n-k-1}$. After division by $n^{n+1}$ the relevant sum is $n^{-2}{\sum }_k{k}^3(n{)}_k/n^k$. Its upper bound follows from $(n{)}_k/n^k\le e^{-k(k-1)/(2n)}$; its lower bound uses $\sqrt{n}\le k\le 2\sqrt{n}$, where this product stays bounded below. The sum is bounded above and below by positive constants. The restriction $n\ge 4$ is necessary: $C(3,4)=0$.