Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$. Then the trees $T_1, T_2, \cdots, T_\ell$ in $G$ are \emph{internally disjoint Steiner trees} connecting $S$ (or $S$-Steiner trees) if $E(T_i) \cap E(T_j )=\emptyset$ and $V(T_i)\cap V(T_j)=S$ for every pair of distinct integers $i,j$, $1 \leq i, j \leq \ell$. Similarly, if we only have the condition $E(T_i) \cap E(T_j )=\emptyset$ but without the condition $V(T_i)\cap V(T_j)=S$, then they are \emph{edge-disjoint Steiner trees}. The \emph{generalized $k$-connectivity}, denoted by $\kappa_k(G)$, of a graph $G$, is defined as $\kappa_k(G)=\min\{\kappa_G(S)|S \subseteq V(G) \ \textrm{and} \ |S|=k \}$, where $\kappa_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees. The \emph{generalized local edge-connectivity} $\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$. The {\it generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is defined as $\lambda_k(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G) \ and \ |S|=k\}$. These measures are generalizations of the concepts of connectivity and edge-connectivity, and they and can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3\leq k\leq \ell^n$, and the exact value of $\kappa_{k}(S(n,\ell))$ for $3\leq k\leq \ell$, where $S(n, \ell)$ is the Sierpi\'{n}ski graphs with order $\ell^n$. As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$. We also study the some network properties of Sierpi\'{n}ski graphs.

翻译：设$G$是一个图，$S\subseteq V(G)$且$|S|\geq 2$。若对于任意一对不同的整数$i,j$（$1 \leq i, j \leq \ell$），树$T_1, T_2, \cdots, T_\ell$满足$E(T_i) \cap E(T_j )=\emptyset$且$V(T_i)\cap V(T_j)=S$，则称这些树为图$G$中连接$S$的**内部不交斯坦纳树**（或$S$-斯坦纳树）。类似地，若仅满足$E(T_i) \cap E(T_j )=\emptyset$而不要求$V(T_i)\cap V(T_j)=S$，则称它们为**边不交斯坦纳树**。图的**广义$k$-连通度**记为$\kappa_k(G)$，定义为$\kappa_k(G)=\min\{\kappa_G(S)|S \subseteq V(G) \ \textrm{且} \ |S|=k \}$，其中$\kappa_G(S)$是内部不交$S$-斯坦纳树的最大数目。**广义局部边连通度**$\lambda_{G}(S)$是图$G$中连接$S$的边不交斯坦纳树的最大数目。图的**广义$k$-边连通度**$\lambda_k(G)$定义为$\lambda_k(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G) \ \textrm{且} \ |S|=k\}$。这些度量是经典连通度和边连通度概念的推广，可作为网络脆弱性的度量指标。通常计算这些广义连通度较为困难，但对某些特殊图类已有精确结果。本文中，我们得到了当$3\leq k\leq \ell^n$时$\lambda_{k}(S(n,\ell))$的精确值，以及当$3\leq k\leq \ell$时$\kappa_{k}(S(n,\ell))$的精确值，其中$S(n, \ell)$是阶数为$\ell^n$的Sierpiński图。作为直接推论，这些图提供了满足$\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$的有趣新实例。此外，我们还研究了Sierpiński图的一些网络性质。