  The conditional diagnosability and the 2-extra connectivity are two important
parameters to measure ability of diagnosing faulty processors and
fault-tolerance in a multiprocessor system. The conditional diagnosability
$t_c(G)$ of $G$ is the maximum number $t$ for which $G$ is conditionally
$t$-diagnosable under the comparison model, while the 2-extra connectivity
$\kappa_2(G)$ of a graph $G$ is the minimum number $k$ for which there is a
vertex-cut $F$ with $|F|=k$ such that every component of $G-F$ has at least $3$
vertices. A quite natural problem is what is the relationship between the
maximum and the minimum problem? This paper partially answer this problem by
proving $t_c(G)=\kappa_2(G)$ for a regular graph $G$ with some acceptable
conditions. As applications, the conditional diagnosability and the 2-extra
connectivity are determined for some well-known classes of vertex-transitive
graphs, including, star graphs, $(n,k)$-star graphs, alternating group
networks, $(n,k)$-arrangement graphs, alternating group graphs, Cayley graphs
obtained from transposition generating trees, bubble-sort graphs, $k$-ary
$n$-cube networks and dual-cubes. Furthermore, many known results about these
networks are obtained directly.
