Let G be a graph with vertex set V = { v1, v2, v3, v4, v5}.
If the degrees of the vertices are 1, 2, 1, 3, 1, respectively, is G a tree? Why or why not?
By definition, a tree is a connected acyaclic graph.
One of the sufficient properties of a tree is that the number of edges equals the number of vertices minus one.
Note: number of edges = (sum of degrees of all vertices)/2.
MathMate: Thank you!
To determine whether G is a tree or not, we need to consider two properties of trees:
1. A tree is a connected graph: It means that there is a path between every pair of vertices in the graph.
2. A tree has no cycles: A cycle is a closed path in which no vertices are repeated, except for the first and last vertices.
Now, for the given information about the degrees of vertices in graph G:
- The degrees of the vertices are 1, 2, 1, 3, 1, respectively.
To determine if G is connected, we can check if there is a path between any two vertices in the graph. In this case, the given degrees do not provide direct information about the existence of paths. So, we need to find a different approach.
Let's analyze the degrees and see if we can identify any inconsistencies:
- If a vertex has degree 1, it means it is connected to only one other vertex.
- If a vertex has degree 2, it means it is connected to exactly two other vertices.
- If a vertex has degree 3, it means it is connected to exactly three other vertices.
Now, let's assign the degrees to the vertices:
v1: degree 1
v2: degree 2
v3: degree 1
v4: degree 3
v5: degree 1
Considering the degrees, we can form the following connections between the vertices:
v1 is connected to v2.
v2 is connected to v1 and v3.
v3 is connected to v2.
v4 is connected to v2, v1, and v5.
v5 is connected to v4.
By examining these connections, we observe that all the vertices are connected, and there are no cycles. This means that G satisfies the properties of a tree. Therefore, G is indeed a tree.