The reliability measure of networks is of significant importance to the design and maintenance of networks. Based on connectivity, many refined quantitative indicators for the reliability of network systems have been introduced. The super vertex edge-connectivity and cyclic edge-connectivity, as important parameters to evaluate the robustness of networks, are explored extensively. As a variant of the hypercube Qn, the varietal hypercube VQn has better properties than Qn with the same number of edges and vertices. Wang and Xu have proved that VQn is super vertex-connected for n≥1 and is also super edge-connected if n≠2. In this paper, we use another method to prove these results. Moreover, we also obtain the super restricted connectivity and the cyclic edge-connectivity of the varietal hypercube VQn.

Reliability evaluation of multiprocessor systems is of significant importance in the design and maintenance of multiprocessor systems. Based on edge-connectivity, more refined quantitative indicators for the reliability of multiprocessor systems have been introduced. The extra edge-connectivity and the component edge-connectivity, as two important parameters to evaluate the robustness of multiprocessor systems, are explored extensively. In this paper, we determine the $h$-extra edge-connectivity and the $(g+1)$-component edge-connectivity of the balanced complete $t$-partite graph $K_{r}^{t}$ for $t, r\geq 2$, where $1\leq h \leq \lfloor tr/2 \rfloor$ and $2\leq g \leq tr-1$.

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