The generalized connectivity of some regular graphs

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is proved to be NP-complete for a general graph $G$. Let $S\subseteq V(G)$ and $\kappa_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots, T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the {\em generalized $k$-connectivity} of a graph $G$ is defined as $\kappa_{k}(G)= min\{\kappa_{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. In this paper, we study the generalized $3$-connectivity of some general $m$-regular and $m$-connected graphs $G_{n}$ constructed recursively and obtain that $\kappa_{3}(G_{n})=m-1$, which attains the upper bound of $\kappa_{3}(G)$ [Discrete Mathematics 310 (2010) 2147-2163] given by Li {\em et al.} for $G=G_{n}$. As applications of the main result, the generalized $3$-connectivity of many famous networks such as the alternating group graph $AG_{n}$, the $k$-ary $n$-cube $Q_{n}^{k}$, the split-star network $S_{n}^{2}$ and the bubble-sort-star graph $BS_{n}$ etc. can be obtained directly.