Prove/Disprove: Let $G$ be a simple graph with $n$ nodes, such that $\forall v\in V: \ deg_{_{G}}(v)\ge k$, then $G$ is a connected graph.
I. $G$ has $7$ nodes, such that $\forall v\in V: deg(v)\ge 2$, prove/ Disprove that $G$ is a connected graph.
II. $G$ has $7$ nodes, such that $\forall v\in V: deg(v)\ge 3$, prove/ Disprove that $G$ is a connected graph.
III. $G$ has $8$ nodes, such that $\forall v\in V: deg(v)\ge 3$, prove/ Disprove that $G$ is a connected graph.
I know that a simple graph is a graph where there are no loops and no parallel edges, and I know that $\\sum_{v\in V} deg(v) \ 2|E|$, but I don't know how to "imagine" a graph as described above in order to find the answer.
Here is a way to envision (i) to see that a graph with seven vertices where each vertex has degree of at least 2 might be disconnected. Try to do the same thing with the other two parts. Either you will have a counterexample or a solid intuition of a proof that the graph must be connected.