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Which of the following statements is/are TRUE for undirected graphs?
P: Number of odd degree vertices is even.
Q: Sum of degrees of all vertices is even.
Q: Sum of degrees of all vertices is even.
- a)P only
- b)Q only
- c)Both P and Q
- d)Neither P nor Q
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Which of the following statements is/are TRUE for undirected graphs?P:...
Q is true: Since the graph is undirected, every edge increases the sum of degrees by 2.
P is true: If we consider sum of degrees and subtract all even degrees, we get an even number (because Q is true). So total number of odd degree vertices must be even.
Most Upvoted Answer
Which of the following statements is/are TRUE for undirected graphs?P:...
Undirected Graphs
Undirected graphs are those graphs in which the edges do not have any direction associated with them. That is, the edges are bidirectional.
Number of Odd Degree Vertices is Even
A vertex in a graph is said to have odd degree if the number of edges incident on the vertex is odd. The sum of degrees of all vertices of an undirected graph counts each edge twice, once for each endpoint. Therefore, the sum of degrees of all vertices in an undirected graph is even.
If we consider the handshaking lemma, it states that the sum of degrees of all vertices in an undirected graph is twice the number of edges. Thus, the sum of degrees of all vertices in an undirected graph is always even.
From the above observations, we can conclude that the number of odd degree vertices in an undirected graph is always even. This statement is true.
Sum of Degrees of all Vertices is Even
As discussed above, the sum of degrees of all vertices in an undirected graph is always even. This statement is true.
Conclusion
Both statements P and Q are true for undirected graphs. Therefore, the correct answer is option C.
Undirected graphs are those graphs in which the edges do not have any direction associated with them. That is, the edges are bidirectional.
Number of Odd Degree Vertices is Even
A vertex in a graph is said to have odd degree if the number of edges incident on the vertex is odd. The sum of degrees of all vertices of an undirected graph counts each edge twice, once for each endpoint. Therefore, the sum of degrees of all vertices in an undirected graph is even.
If we consider the handshaking lemma, it states that the sum of degrees of all vertices in an undirected graph is twice the number of edges. Thus, the sum of degrees of all vertices in an undirected graph is always even.
From the above observations, we can conclude that the number of odd degree vertices in an undirected graph is always even. This statement is true.
Sum of Degrees of all Vertices is Even
As discussed above, the sum of degrees of all vertices in an undirected graph is always even. This statement is true.
Conclusion
Both statements P and Q are true for undirected graphs. Therefore, the correct answer is option C.
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Which of the following statements is/are TRUE for undirected graphs?P: Number of odd degree vertices is even.Q: Sum of degrees of all vertices is even.a)P onlyb)Q onlyc)Both P and Qd)Neither P nor QCorrect answer is option 'C'. Can you explain this answer?
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