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Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G − S is disconnected and each component of G − S has at least h + 1 vertices. The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The h-extra connectivity for h = 1, 2 of k-ary n-cube are gotten by Hsieh et al. in [Theoretical Computer Science, 443 (2012) 63-69] for k ≥ 4 and Zhu et al. in [Theory of Computing Systems, arxiv.org/pdf/1105.0991v1 [cs.DM] 5May 2011] for k = 3. In this paper, we show that the h-extra connectivity of the 3-ary n-cube networks for h = 3 is equal to 8n − 12, where n ≥ 3.

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Extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system's ability to tolerate and diagnose faulty processor. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty vertices within a set containing at most one fault-free vertex. In this paper, the result that the pessimistic diagnosability t p (G) equals the extra connectivity κ 1 (G) of a regular graph G under some conditions are shown. Furthermore, the following new results are gotten: the pessimistic diagnosability t p (S 2 n ) = 4n − 9 for split-star networks S 2 n ; t p (Γ n ) = 2n − 4 for Cayley graphs generated by transposition trees Γ n ; t p (Γ n (∆)) = 4n−11 for Cayley graph generated by the 2-tree Γ n (∆); t p (BP n ) = 2n−2 for the burnt pancake networks BP n . As corollaries, the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs; the alternating group networks; BC networks; the k-ary n-cube networks etc. are obtained directly.

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