Graphs with large generalized (edge-)connectivity

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized $k$-edge-connectivity $\lambda_k(G)$. In this paper, graphs of order $n$ such that $\kappa_k(G)=n-\frac{k}{2}-1$ and $\lambda_k(G)=n-\frac{k}{2}-1$ for even $k$ are characterized.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1207.183

The generalized 3-connectivity of Lexicographic product graphs

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs $G$ and $H$, $\kappa_3(G\circ H)\geq \kappa_3(G)|V(H)|$. We also give upper bounds for $\kappa_3(G\Box H)$ and $\kappa_3(G\circ H)$. Moreover, all the bounds are sharp.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1103.609