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- W2889006091 abstract "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 $Ssubseteq V(G)$ 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 $ineq j$. For an integer $k$ with $2leq kleq n$, the {em generalized $k$-connectivity} of a graph $G$ is defined as $kappa_{k}(G)= min{kappa_{G}(S)|Ssubseteq V(G)$ 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." @default.
- W2889006091 created "2018-09-07" @default.
- W2889006091 creator A5022507152 @default.
- W2889006091 creator A5049116158 @default.
- W2889006091 date "2018-08-29" @default.
- W2889006091 modified "2023-09-25" @default.
- W2889006091 title "The generalized connectivity of some regular graphs" @default.
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- W2889006091 doi "https://doi.org/10.48550/arxiv.1808.10074" @default.
- W2889006091 hasPublicationYear "2018" @default.
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