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If a simple graph G, contains n vertices and m edges, the number of edges in the Graph G'(Complement of G) is ___________
- a)(n*n-n-2*m)/2
- b)(n*n+n+2*m)/2
- c)(n*n+n-2*m)/2
- d)(n*n-n+2*m)/2
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If a simple graph G, contains n vertices and m edges, the number of ed...
The union of G and G’ would be a complete graph so, the number of edges in G’= number of edges in the complete form of G(nC2)-edges in G(m).
Most Upvoted Answer
If a simple graph G, contains n vertices and m edges, the number of ed...
The complement of a graph G, denoted as G', is a graph that contains all the vertices of G, but does not have any of the edges that are present in G. In other words, if there is an edge between two vertices in G, there is no edge between those vertices in G'.
To find the number of edges in G', we need to consider the number of possible edges that can be formed between the vertices of G, and subtract the number of edges that are already present in G.
Let's break down the problem into steps to understand the solution better:
Step 1: Counting the total number of possible edges
In a simple graph G with n vertices, the total number of possible edges is given by the formula n(n-1)/2. This formula is derived from the fact that each vertex can be connected to (n-1) other vertices, but we divide by 2 to avoid counting each edge twice.
Step 2: Counting the number of edges in G
The given graph G contains m edges. So, the number of edges in G is m.
Step 3: Finding the number of edges in G'
To find the number of edges in G', we subtract the number of edges in G from the total number of possible edges. Mathematically, this can be represented as:
Number of edges in G' = Total number of possible edges - Number of edges in G
Substituting the formulas from Step 1 and Step 2, we get:
Number of edges in G' = n(n-1)/2 - m
Simplifying the expression further, we have:
Number of edges in G' = (n^2 - n - 2m)/2
Therefore, the correct answer is option A: (n^2 - n - 2m)/2.
To find the number of edges in G', we need to consider the number of possible edges that can be formed between the vertices of G, and subtract the number of edges that are already present in G.
Let's break down the problem into steps to understand the solution better:
Step 1: Counting the total number of possible edges
In a simple graph G with n vertices, the total number of possible edges is given by the formula n(n-1)/2. This formula is derived from the fact that each vertex can be connected to (n-1) other vertices, but we divide by 2 to avoid counting each edge twice.
Step 2: Counting the number of edges in G
The given graph G contains m edges. So, the number of edges in G is m.
Step 3: Finding the number of edges in G'
To find the number of edges in G', we subtract the number of edges in G from the total number of possible edges. Mathematically, this can be represented as:
Number of edges in G' = Total number of possible edges - Number of edges in G
Substituting the formulas from Step 1 and Step 2, we get:
Number of edges in G' = n(n-1)/2 - m
Simplifying the expression further, we have:
Number of edges in G' = (n^2 - n - 2m)/2
Therefore, the correct answer is option A: (n^2 - n - 2m)/2.
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