Connectivity plays an important role in measuring the fault tolerance of interconnection networks $G=(V,E)$. A faulty set $F⊆V$ is called a g-extra faulty set if every component of G−F has more than g nodes. A g-extra cut of G is a g-extra faulty set F such that G−F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. Diagnosability is an important metric for measuring the reliability of G. A new measure for fault diagnosis of G restrains that every fault-free component has at least $(g+1)$ fault-free nodes, which is called the g-extra diagnosability of G. As a favorable topology structure of interconnection networks, the n-dimensional bubble-sort star graph BS_n has many good properties. In this paper, we prove that 2-extra connectivity of BS_n is $6n−15$ for $n≥5$ and the 2-extra connectivity of BS₄ is 8; the 2-extra diagnosability of BS_n is $6n−13$ under the PMC model for $n≥5$ and the 2-extra diagnosability of BS_n is $6n−13$ under the MM^* model for $n≥6$.

Connectivity and Diagnosability play an important role in measuring the fault tolerance of interconnection networks. As a topology structure of interconnection networks, the expanded k-ary n-cube X Q k n has many good properties. In this paper, we prove that (1) the connectivity of X Q k n is 4n; (2) the nature connectivity of X Q k n is 8n − 4; (3) the nature diagnosability of X Q k n under the PMC model and MM∗ model is 8n − 3 for n ≥ 2.