Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, Fàbrega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq κ^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{Δ(G),\left\lfloor \frac{n-3}{2}\right\rfloor\right\}$, and graphs with $κ^g(G)=1,2$ and trees with $κ^g(T_n)=n-t$ 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.