  Connectivity and diagnosability are two important parameters for the fault
tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol
proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that
$1\leq \kappa^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{\Delta(G),\left\lfloor
\frac{n-3}{2}\right\rfloor\right\}$, and graphs with $\kappa^g(G)=1,2$ and
trees with $\kappa^g(T_n)=n-t$ for $4\leq t\leq \frac{n+2}{2}$ are
characterized, respectively. In the end, we get the three extremal results for
the $g$-good-neighbor connectivity.
