Which condition is necessarily for a graph to be connected?

Key idea: A simple undirected graph on n vertices that has more than C(n-1,2) = (n-1)(n-2)/2 edges must be connected. The reason is that a disconnected graph can have at most C(n-1,2) edges (take a component of size n-1 and an isolated vertex).

1. Graph with 6 vertices and 10 edges: For n = 6, C(5,2) = 10. Since 10 is not greater than 10, this does not guarantee connectivity. Example: a complete graph on 5 vertices (10 edges) plus one isolated vertex is disconnected.

2. Graph with 7 vertices and 14 edges: For n = 7, C(6,2) = 15. Because 14 ≤ 15, connectivity is not guaranteed. A disconnected arrangement is possible (for example, a 6-vertex component with 14 edges plus an isolated vertex).

3. Graph with 8 vertices and 22 edges: For n = 8, C(7,2) = 21. Because 22 > 21, the graph must be connected. No disconnected graph on 8 vertices can have more than 21 edges.

4. Graph with 9 vertices and 28 edges: For n = 9, C(8,2) = 28. Since 28 is not greater than 28, this does not guarantee connectivity. Example: a complete graph on 8 vertices (28 edges) plus one isolated vertex is disconnected.

Conclusion: Among the given choices, the graph with 8 vertices and 22 edges is necessarily connected because it exceeds the C(n-1,2) threshold.