Final answer:
The degree of vertex 8 should be 9 based on the Handshaking Lemma and the given degrees of other vertices. However, since 9 is not an option and all odd-numbered vertices have a degree of 8, option (b) 8 is the best guess.
option b is the correct
Step-by-step explanation:
To determine the degree of vertex 8 in the undirected graph with vertices labeled 1 through 8, we can use the information provided about the degrees of the other vertices. The sum of the degrees of all vertices in an undirected graph is equal to twice the number of edges, known as the Handshaking Lemma.
Given that there are 31 edges, the sum of the degrees of all vertices would be 2 × 31 = 62. We are told that vertices 1, 3, 5, and 7 all have degree 8 and that vertices 2, 4, and 6 have degree 7. Therefore, the combined degrees of vertices 1, 3, 5, and 7 is 4 × 8 = 32, and the combined degrees of vertices 2, 4, and 6 is 3 × 7 = 21. Adding these together gives us 32 + 21 = 53.
To find the degree of vertex 8, we subtract the sum of the known degrees from the total degree count:
62 - 53 = 9
Thus, the degree of vertex 8 is 9. However, this is not an option in the question, and there seems to be a discrepancy. Since the closest degree value provided in the options and doesn't exceed the total is 8 and matches the pattern of odd-numbered vertices having a higher degree, it's possible that there might be a typo in the question or the information given. Based on the calculations, option (b) 8 would be the best guess under the assumption that there may be a mistake in the provided details.