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 $i, j \in \{1, 2, \cdots , r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the 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 introduce a family of regular graph $G_{n}$ that can be constructed recursively and each vertex with exactly one outside neighbor. The generalized $3$-connectivity of the regular graph $G_{n}$ is studied, which attains a previously proven upper bound on $\kappa _{3}(G)$. As applications of the main result, the generalized $3$-connectivity of some important networks including some known results such as the alternating group network $AN_{n}$, the star graph $S_{n}$ and the pancake graphs $P_{n}$ can be obtained directly.