My questions/comments are bolded:

What are connected components?

6. Following is the algorithm to find the connected components of a graph G.

for i = 1 to n

if i is not yet explored

BFS(G, i)   -- run the BFS search algorithm on node i of graph G

The runtime for this algorithm is

 O(n^2) because we run the BFS algorithm, which is O(n), once for each node of the graph
 O(n) because we run the BFS algorithm, which is O(n), once for each node that wasn't touched by a previous BFS run
 O(n^2) because we run the BFS algorithm, which is O(n^2), once for each node that wasn't touched by a previous BFS run
 O(n) because we run the BFS algorithm, which is O(1), once for each node of the graph
 
 I got this exercise wrong. But I assume that it is either the first or third option due to the connected component's algorithm. The BFS has a while loop which is n and an inner for loop which is another n, yielding n^2. My guess leans more towards option 3.

rabbott wrote:

The answer is 

O(n) because we run the BFS algorithm, which is O(n), once for each node that wasn't touched by a previous BFS run

This algorithm simply touches each node once. Hence it's O(n). The only reason the last answer is wrong is that it claims that BFS is O(1).

I selected the last answer on my quiz. It definitely threw off my reasoning for another attempt to answer this question.

rabbott wrote:

Actually, the last answer is wrong also because it claims that BFS is run for each node. It touches each node, but it's not run separately for each node. 

Makes sense, thanks.

See https://en.wikipedia.org/wiki/Connected_component_%28graph_theory%29