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What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of degree 2, 3 vertices of degree 4 and remaining of degree 3?
- a)10
- b)11
- c)18
- d)19
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
What is the number of vertices in an undirected connected graph with 2...
Sum of degree of all vertices = 2e (using Handshaking lemma)
So, 6 x 2 + 3 x 4 + x x 3 = 2 x 27
12 + 12 + 3x = 54
3x = 54 - 24
x = 30/3
x = 10
So, total number of vertices
⇒ x + 6 + 3 = 10 + 6 + 3 = 19
So, 6 x 2 + 3 x 4 + x x 3 = 2 x 27
12 + 12 + 3x = 54
3x = 54 - 24
x = 30/3
x = 10
So, total number of vertices
⇒ x + 6 + 3 = 10 + 6 + 3 = 19
Most Upvoted Answer
What is the number of vertices in an undirected connected graph with 2...
To find the number of vertices in the undirected connected graph, we can use the Handshaking Lemma. The Handshaking Lemma states that in any undirected graph, the sum of the degrees of all vertices is twice the number of edges.
Let's calculate the sum of the degrees of all vertices in the given graph:
- The 6 vertices of degree 2 contribute a total of 6*2 = 12 degrees.
- The 3 vertices of degree 4 contribute a total of 3*4 = 12 degrees.
- The remaining vertices of degree 3 contribute a total of (n-6-3)*3 = (n-9)*3 degrees, where n is the total number of vertices in the graph.
Now, according to the Handshaking Lemma, the sum of the degrees of all vertices is twice the number of edges, which is 27. Therefore, we have:
12 + 12 + (n-9)*3 = 2*27
24 + 3n - 27 = 54
3n - 3 = 54 - 24
3n - 3 = 30
3n = 33
n = 11
The number of vertices in the undirected connected graph is 11, which corresponds to option D.
Let's calculate the sum of the degrees of all vertices in the given graph:
- The 6 vertices of degree 2 contribute a total of 6*2 = 12 degrees.
- The 3 vertices of degree 4 contribute a total of 3*4 = 12 degrees.
- The remaining vertices of degree 3 contribute a total of (n-6-3)*3 = (n-9)*3 degrees, where n is the total number of vertices in the graph.
Now, according to the Handshaking Lemma, the sum of the degrees of all vertices is twice the number of edges, which is 27. Therefore, we have:
12 + 12 + (n-9)*3 = 2*27
24 + 3n - 27 = 54
3n - 3 = 54 - 24
3n - 3 = 30
3n = 33
n = 11
The number of vertices in the undirected connected graph is 11, which corresponds to option D.
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What is the number of vertices in an undirected connected graph with 27 edges, 6 vertices of degree 2, 3 vertices of degree 4 and remaining of degree 3?a)10b)11c)18d)19Correct answer is option 'D'. Can you explain this answer?
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