The minimal size of a graph with given generalized 3-edge-connectivity

For $S\subseteq V(G)$ and $|S|\geq 2$, $\lambda(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is then defined as $\lambda_k(G)= min\{\lambda(S) : S\subseteq V(G) \ and \ |S|=k\}$. It is also clear that when $|S|=2$, $\lambda_2(G)$ is nothing new but the standard edge-connectivity $\lambda(G)$ of $G$. In this paper, graphs of order $n$ such that $\lambda_3(G)=n-3$ is characterized. Furthermore, we determine the minimal number of edges of a graph of order $n$ with $\lambda_3=1,n-3,n-2$ and give a sharp lower bound for $2\leq \lambda_3\leq n-4$.