Some basic properties of independence number of graph

Question

Suppose that $G=(V,E)$ is a simple graph on $n$ vertices and consider $\alpha(G)$ - independence number of $G$.

I have some questions related with quantity $\alpha(G)$.

1. Am I right that $\alpha(G)\neq0$ because for any $v\in V$ you can consider $\{v\}$ as an independent set, right?

2. If $\alpha(G)=1$, then $G$ is a complete graph on $n$ vertices, right?