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An undirected graph G with only one simple path between each pair of vertices has two vertices of degree 4, one vertex of degree 3 and two vertices of degree 2.
Number of vertices of degree 1 are ______________
Correct answer is '7'. Can you explain this answer?
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An undirected graph G with only one simple path between each pair of v...
Concept:
Since there is only one simple path between each pair of vertices, the given graph is a tree.
Tree with n nodes has (n -1) edges
Let the number of vertices of degree 1 be x.
2 vertices have degree 4
2 vertices have degree 2
1 vertex has degree 3
Since there is only one simple path between each pair of vertices, the given graph is a tree.
Tree with n nodes has (n -1) edges
Let the number of vertices of degree 1 be x.
2 vertices have degree 4
2 vertices have degree 2
1 vertex has degree 3
Calculation:
Total number of nodes in a graph = x + 2 + 2 + 1 = x + 5
Total number of edges in a graph = x + 5 - 1 = x + 4
According to Handshaking Lemma,
∑deg(v) = 2|E| where E is Edges.
2×4+1×3+2×2+x=2(x+4)
15 + x = 8 + 2x
∴ x = 7
Total number of nodes in a graph = x + 2 + 2 + 1 = x + 5
Total number of edges in a graph = x + 5 - 1 = x + 4
According to Handshaking Lemma,
∑deg(v) = 2|E| where E is Edges.
2×4+1×3+2×2+x=2(x+4)
15 + x = 8 + 2x
∴ x = 7
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Community Answer
An undirected graph G with only one simple path between each pair of v...
The given information describes an undirected graph with specific degrees of vertices. We need to determine the number of vertices with degree 1.
Understanding the Graph
An undirected graph consists of vertices (nodes) and edges (connections between vertices). In this case, the graph has certain degree values for each vertex, representing the number of edges connected to that vertex.
The given graph has:
- 2 vertices with degree 4
- 1 vertex with degree 3
- 2 vertices with degree 2
Degree of a Vertex
The degree of a vertex is the count of edges connected to it. In an undirected graph, the degree is the same for both incoming and outgoing edges.
Simple Path
A simple path is a path in a graph that does not contain any repeated vertices. In other words, it is a path that visits each vertex only once.
Analysis
Let's analyze the information given and try to determine the number of vertices with degree 1.
Vertices with Degree 4
Since there are two vertices with degree 4, let's label them as A and B. The only simple path between these two vertices can be AB or BA.
Vertex with Degree 3
There is one vertex with degree 3, let's label it as C.
Vertices with Degree 2
Since there are two vertices with degree 2, let's label them as D and E.
Connecting the Vertices
Based on the given information, we can connect the vertices as follows:
- A is connected to B (degree 4)
- B is connected to C (degree 3)
- C is connected to D (degree 2)
- D is connected to E (degree 2)
Calculating the Remaining Degrees
To determine the number of vertices with degree 1, we need to calculate the remaining degrees.
- Degree of A: 4 (already determined)
- Degree of B: 4 (already determined)
- Degree of C: 3 (already determined)
- Degree of D: 2 (already determined)
- Degree of E: 2 (already determined)
To find the remaining degrees, we can use the handshaking lemma, which states that the sum of degrees of all vertices in an undirected graph is twice the number of edges.
We have the following vertices and degrees:
- A: 4
- B: 4
- C: 3
- D: 2
- E: 2
The total sum of degrees is 4 + 4 + 3 + 2 + 2 = 15.
Using the handshaking lemma, we can calculate the number of edges:
2 * number of edges = sum of degrees
2 * number of edges = 15
number of edges = 15 / 2
number of edges = 7.5
Since the graph is described as having a simple path between each pair of vertices, the number of edges should be a whole number. Therefore, there must be 7 edges in the graph.
Calculating the Degree 1 Vertices
To calculate the number of vertices with degree 1, we can subtract the degrees of the given vertices from the total sum of degrees in the graph.
Total sum
Understanding the Graph
An undirected graph consists of vertices (nodes) and edges (connections between vertices). In this case, the graph has certain degree values for each vertex, representing the number of edges connected to that vertex.
The given graph has:
- 2 vertices with degree 4
- 1 vertex with degree 3
- 2 vertices with degree 2
Degree of a Vertex
The degree of a vertex is the count of edges connected to it. In an undirected graph, the degree is the same for both incoming and outgoing edges.
Simple Path
A simple path is a path in a graph that does not contain any repeated vertices. In other words, it is a path that visits each vertex only once.
Analysis
Let's analyze the information given and try to determine the number of vertices with degree 1.
Vertices with Degree 4
Since there are two vertices with degree 4, let's label them as A and B. The only simple path between these two vertices can be AB or BA.
Vertex with Degree 3
There is one vertex with degree 3, let's label it as C.
Vertices with Degree 2
Since there are two vertices with degree 2, let's label them as D and E.
Connecting the Vertices
Based on the given information, we can connect the vertices as follows:
- A is connected to B (degree 4)
- B is connected to C (degree 3)
- C is connected to D (degree 2)
- D is connected to E (degree 2)
Calculating the Remaining Degrees
To determine the number of vertices with degree 1, we need to calculate the remaining degrees.
- Degree of A: 4 (already determined)
- Degree of B: 4 (already determined)
- Degree of C: 3 (already determined)
- Degree of D: 2 (already determined)
- Degree of E: 2 (already determined)
To find the remaining degrees, we can use the handshaking lemma, which states that the sum of degrees of all vertices in an undirected graph is twice the number of edges.
We have the following vertices and degrees:
- A: 4
- B: 4
- C: 3
- D: 2
- E: 2
The total sum of degrees is 4 + 4 + 3 + 2 + 2 = 15.
Using the handshaking lemma, we can calculate the number of edges:
2 * number of edges = sum of degrees
2 * number of edges = 15
number of edges = 15 / 2
number of edges = 7.5
Since the graph is described as having a simple path between each pair of vertices, the number of edges should be a whole number. Therefore, there must be 7 edges in the graph.
Calculating the Degree 1 Vertices
To calculate the number of vertices with degree 1, we can subtract the degrees of the given vertices from the total sum of degrees in the graph.
Total sum
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An undirected graph G with only one simple path between each pair of vertices has two vertices of degree 4, one vertex of degree 3 and two vertices of degree 2.Number of vertices of degree 1 are ______________Correct answer is '7'. Can you explain this answer?
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