Let G be a simple graph with 10 connected components, such that each connected component contains at least 7 edges. What is the minimum number of vertices G can have? A. 52 B. 49 C. 51 D. 50
Question
Let G be a simple graph with 10 connected components, such that each connected component contains at least 7 edges. What is the minimum number of vertices G can have? A. 52
B. 49
C. 51
D. 50
Solution
The minimum number of vertices in a connected component with 7 edges is 4. This is because a tree (a connected graph with no cycles) has one less edge than it has vertices, and adding one edge to a tree creates a single cycle, which is the minimum for a connected graph. Therefore, each connected component must have at least 4 vertices. Since there are 10 connected components, the graph G must have at least 4*10 = 40 vertices. So, none of the options A, B, C, D is correct.
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