Reliability Assessment of Some Regular Networks

The generalized $k$-connectivity 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 a generalization of traditional connectivity. 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, ...