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For $S\subseteq V(G)$ and $|S|\geq 2$, $\lambda(S)$ is the maximum number ofedge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leqn$, the \emph{generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is thendefined 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 thestandard edge-connectivity $\lambda(G)$ of $G$. In this paper, graphs of order$n$ such that $\lambda_3(G)=n-3$ is characterized. Furthermore, we determinethe 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 $2\leq \lambda_3\leq n-4$.

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