Consider the following statements about simple connected undirected g...
As the graph is simple, there is no self loop or parallel edge. As the graph is connected, no vertex can have 0 degree and hence, the degree range from 1 to n-1 which implies that the degree of at least two vertices must be the same.
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Consider the following statements about simple connected undirected g...
Explanation:
1. Statement 1: At least two vertices have the same degree
In a simple connected undirected graph with more than 2 vertices, the degree of a vertex can range from 0 (isolated vertex) to n-1 (complete graph). However, it is not necessary that at least two vertices have the same degree. For example, consider a path graph with vertices having degrees 1 and 2 respectively. In this case, no two vertices have the same degree.
2. Statement 2: At least three vertices have the same degree
In a simple connected undirected graph with more than 2 vertices, the Pigeonhole Principle can be applied to show that at least three vertices have the same degree. Since there are n-2 possible degrees (ranging from 1 to n-1) and n-2 vertices (excluding the endpoints of the path), there must be at least one degree that is shared by at least three vertices. This is because if each degree is unique to two or fewer vertices, the sum of the degrees would exceed 2(n-2), which is not possible in a simple graph.
Therefore, statement 1 is not always true, but statement 2 holds true in any simple connected undirected graph with more than 2 vertices. Hence, the correct answer is option 'B' - 1 only.