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Which one of the following is TRUE for any simple connected undirected graph with more than 2 vertices?
- a)No two vertices have the same degree.
- b)At least two vertices have the same degree.
- c)At least three vertices have the same degree.
- d)All vertices have the same degree.
Correct answer is option 'B'. Can you explain this answer?
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Which one of the following is TRUE for any simple connected undirected...
Since the graph is simple, there must not be any self loop and parallel edges. Since the graph is connected, the degree of any vertex cannot be 0. Therefore, degree of all vertices should be be from 1 to n-1. So the degree of at least two vertices must be same.
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Which one of the following is TRUE for any simple connected undirected...
Explanation:
Simple connected undirected graph: A simple graph is a graph that does not have multiple edges or self-loops. A connected graph is a graph in which there is a path between any two vertices. An undirected graph is a graph in which edges do not have a direction.
In a graph, the degree of a vertex is the number of edges incident to it.
To determine the true statement for any simple connected undirected graph with more than 2 vertices, let's analyze each option:
a) No two vertices have the same degree: This statement is not necessarily true. In a graph, it is possible for two vertices to have the same degree. For example, in a cycle graph, all vertices have a degree of 2.
b) At least two vertices have the same degree: This statement is true. Let's assume the graph has n vertices. The minimum degree of a vertex in a connected graph is 1, and the maximum degree is n-1. If all vertices have different degrees, then the possible degrees for the n vertices would be 1, 2, 3, ..., n-1. But there are only n-1 possible degrees. Therefore, there must be at least two vertices with the same degree.
c) At least three vertices have the same degree: This statement is not necessarily true. It is possible for two vertices to have the same degree, but there is no guarantee that three vertices will have the same degree.
d) All vertices have the same degree: This statement is not necessarily true. In a connected graph, unless the graph is a complete graph, it is not possible for all vertices to have the same degree.
Therefore, the correct answer is option B: At least two vertices have the same degree.
Simple connected undirected graph: A simple graph is a graph that does not have multiple edges or self-loops. A connected graph is a graph in which there is a path between any two vertices. An undirected graph is a graph in which edges do not have a direction.
In a graph, the degree of a vertex is the number of edges incident to it.
To determine the true statement for any simple connected undirected graph with more than 2 vertices, let's analyze each option:
a) No two vertices have the same degree: This statement is not necessarily true. In a graph, it is possible for two vertices to have the same degree. For example, in a cycle graph, all vertices have a degree of 2.
b) At least two vertices have the same degree: This statement is true. Let's assume the graph has n vertices. The minimum degree of a vertex in a connected graph is 1, and the maximum degree is n-1. If all vertices have different degrees, then the possible degrees for the n vertices would be 1, 2, 3, ..., n-1. But there are only n-1 possible degrees. Therefore, there must be at least two vertices with the same degree.
c) At least three vertices have the same degree: This statement is not necessarily true. It is possible for two vertices to have the same degree, but there is no guarantee that three vertices will have the same degree.
d) All vertices have the same degree: This statement is not necessarily true. In a connected graph, unless the graph is a complete graph, it is not possible for all vertices to have the same degree.
Therefore, the correct answer is option B: At least two vertices have the same degree.
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