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We study the generalized $k$-connectivity ${\kappa }_k\left(G\right)$ as introduced by Hager in 1985, as well as the more recently introduced generalized $k$-edge-connectivity ${\lambda }_k\left(G\right)$. We determine the exact value of ${\kappa }_k\left(G\right)$ and ${\lambda }_k\left(G\right)$ for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case $k=3$.