A simple graph (a graph without parallel edge or loops) with n vertice...
A graph with n-edges (without loop and parallel edges) and K components can have
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A simple graph (a graph without parallel edge or loops) with n vertice...
Explanation:
To understand why the correct answer is option A, let's break down the different options and analyze them one by one.
Option A: n edges
- This option states that a simple graph with n vertices and k components can have at most n edges.
- In a simple graph, each edge connects two distinct vertices. So, the maximum number of edges in a graph with n vertices is n.
- Therefore, this option is correct as it represents the maximum number of edges in a simple graph.
Option B: n - k edges
- This option states that a simple graph with n vertices and k components can have at most n - k edges.
- However, this is not true as it implies that each component in the graph can have at most one edge.
- In reality, each component can have multiple edges, depending on the connections between the vertices within the component.
- Therefore, this option is incorrect.
Option C: (n - k)(n - k + 1)
- This option states that a simple graph with n vertices and k components can have at most (n - k)(n - k + 1) edges.
- However, this formula does not accurately represent the maximum number of edges in a simple graph.
- The correct formula for the maximum number of edges in a simple graph with n vertices is n(n - 1)/2, which is derived from the fact that each vertex can be connected to every other vertex except itself and the vertices it is already connected to.
- Therefore, this option is incorrect.
Option D: (n - k)(n - k + 1)/2 edges
- This option is similar to option C but divides the result by 2.
- As mentioned earlier, the correct formula for the maximum number of edges in a simple graph with n vertices is n(n - 1)/2.
- Therefore, this option is also incorrect.
Conclusion:
- After analyzing all the options, we can conclude that option A is the correct answer.
- A simple graph with n vertices and k components can have at most n edges, as each vertex can be connected to every other vertex except itself.
- This is the maximum number of edges possible in a simple graph, and any additional edges would result in parallel edges or loops, which are not allowed in a simple graph.