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. [ABSTRACT FROM AUTHOR]