  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 $\kappa_{1}(G)$ of a
regular graph $G$ under some conditions are shown. Furthermore, the following
new results are gotten: the pessimistic diagnosability $t_p(S_n^2)=4n-9$ for
split-star networks $S_n^2$, $t_p(\Gamma_n)=2n-4$ for Cayley graphs generated
by transposition trees $\Gamma_n$, $t_p(\Gamma_{n}(\Delta))=4n-11$ for Cayley
graph generated by the $2$-tree $\Gamma_{n}(\Delta)$, $t_{p}(BP_n)=2n-2$ 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.
