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The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?
I. 7, 6, 5, 4, 4, 3, 2, 1
II. 6, 6, 6, 6, 3, 3, 2, 2
III. 7, 6, 6, 4, 4, 3, 2, 2
IV. 8, 7, 7, 6, 4, 2, 1, 1
I. 7, 6, 5, 4, 4, 3, 2, 1
II. 6, 6, 6, 6, 3, 3, 2, 2
III. 7, 6, 6, 4, 4, 3, 2, 2
IV. 8, 7, 7, 6, 4, 2, 1, 1
- a)I and II
- b)III and IV
- c)IV only
- d)II and IV
Correct answer is option 'D'. Can you explain this answer?
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The degree sequence of a simple graph is the sequence of the degrees o...
Explanation:
To determine whether a given sequence can be a degree sequence of a graph, we need to check if it satisfies the Handshaking Lemma, which states that the sum of the degrees of all nodes in a graph is equal to twice the number of edges.
I. 7, 6, 5, 4, 4, 3, 2, 1
- The sum of the degrees is 7 + 6 + 5 + 4 + 4 + 3 + 2 + 1 = 32.
- The number of edges should be half of the sum of the degrees, which is 32 / 2 = 16.
- It is possible to construct a graph with this degree sequence. For example, a complete graph with 8 nodes will have this degree sequence.
II. 6, 6, 6, 6, 3, 3, 2, 2
- The sum of the degrees is 6 + 6 + 6 + 6 + 3 + 3 + 2 + 2 = 34.
- The number of edges should be half of the sum of the degrees, which is 34 / 2 = 17.
- It is not possible to construct a graph with this degree sequence since the sum of degrees is odd, but the number of edges must be an integer.
III. 7, 6, 6, 4, 4, 3, 2, 2
- The sum of the degrees is 7 + 6 + 6 + 4 + 4 + 3 + 2 + 2 = 34.
- The number of edges should be half of the sum of the degrees, which is 34 / 2 = 17.
- It is possible to construct a graph with this degree sequence. For example, a graph with 8 nodes where one node has degree 7, two nodes have degree 6, two nodes have degree 4, and the remaining nodes have degrees 3, 2, and 2.
IV. 8, 7, 7, 6, 4, 2, 1, 1
- The sum of the degrees is 8 + 7 + 7 + 6 + 4 + 2 + 1 + 1 = 36.
- The number of edges should be half of the sum of the degrees, which is 36 / 2 = 18.
- It is not possible to construct a graph with this degree sequence since the sum of degrees is even, but the number of edges must be half of an even number.
Conclusion:
The sequences that can not be the degree sequence of any graph are II. 6, 6, 6, 6, 3, 3, 2, 2 and IV. 8, 7, 7, 6, 4, 2, 1, 1. Therefore, the correct answer is option 'D'.
To determine whether a given sequence can be a degree sequence of a graph, we need to check if it satisfies the Handshaking Lemma, which states that the sum of the degrees of all nodes in a graph is equal to twice the number of edges.
I. 7, 6, 5, 4, 4, 3, 2, 1
- The sum of the degrees is 7 + 6 + 5 + 4 + 4 + 3 + 2 + 1 = 32.
- The number of edges should be half of the sum of the degrees, which is 32 / 2 = 16.
- It is possible to construct a graph with this degree sequence. For example, a complete graph with 8 nodes will have this degree sequence.
II. 6, 6, 6, 6, 3, 3, 2, 2
- The sum of the degrees is 6 + 6 + 6 + 6 + 3 + 3 + 2 + 2 = 34.
- The number of edges should be half of the sum of the degrees, which is 34 / 2 = 17.
- It is not possible to construct a graph with this degree sequence since the sum of degrees is odd, but the number of edges must be an integer.
III. 7, 6, 6, 4, 4, 3, 2, 2
- The sum of the degrees is 7 + 6 + 6 + 4 + 4 + 3 + 2 + 2 = 34.
- The number of edges should be half of the sum of the degrees, which is 34 / 2 = 17.
- It is possible to construct a graph with this degree sequence. For example, a graph with 8 nodes where one node has degree 7, two nodes have degree 6, two nodes have degree 4, and the remaining nodes have degrees 3, 2, and 2.
IV. 8, 7, 7, 6, 4, 2, 1, 1
- The sum of the degrees is 8 + 7 + 7 + 6 + 4 + 2 + 1 + 1 = 36.
- The number of edges should be half of the sum of the degrees, which is 36 / 2 = 18.
- It is not possible to construct a graph with this degree sequence since the sum of degrees is even, but the number of edges must be half of an even number.
Conclusion:
The sequences that can not be the degree sequence of any graph are II. 6, 6, 6, 6, 3, 3, 2, 2 and IV. 8, 7, 7, 6, 4, 2, 1, 1. Therefore, the correct answer is option 'D'.
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The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer?
Question Description
The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer?.
The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The degree sequence of a simple graph is the sequence of the degrees of the nodes in the graph in decreasing order. Which of the following sequences can not be the degree sequence of any graph?I. 7, 6, 5, 4, 4, 3, 2, 1II. 6, 6, 6, 6, 3, 3, 2, 2III. 7, 6, 6, 4, 4, 3, 2, 2IV. 8, 7, 7, 6, 4, 2, 1, 1a)I and IIb)III and IVc)IV onlyd)II and IVCorrect answer is option 'D'. Can you explain this answer?.
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