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What is the sum of the degree of each vertex for an undirected graph with m vertices and n edges ?
- a)(2n - 1)/2
- b)mn
- c)2n
- d)2m
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
What is the sum of the degree of each vertex for an undirected graph w...
Concept:
A tree with n vertices has e edges
The Handshaking Theorem states that the sum of the degrees of all the vertices of G is twice the number of edges in G.
A tree with n vertices has e edges
The Handshaking Theorem states that the sum of the degrees of all the vertices of G is twice the number of edges in G.
Formula:
∑ di = 2 × e (By Handshaking Lemma)
∑ di = 2 × e (By Handshaking Lemma)
Explanation:
Given an undirected graph total of n edges
e=n
and ∑ di is the sum of the degree of each vertex
∑ di = 2 × e
∑ di = 2n
So, the sum of the degree of each vertex is 2n
Given an undirected graph total of n edges
e=n
and ∑ di is the sum of the degree of each vertex
∑ di = 2 × e
∑ di = 2n
So, the sum of the degree of each vertex is 2n
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Community Answer
What is the sum of the degree of each vertex for an undirected graph w...
The sum of the degrees of all the vertices in an undirected graph can be calculated by counting the number of edges incident to each vertex. Each edge contributes 2 to the sum of the degrees since it is counted once for each vertex it is incident to.
Let's break down the problem step by step:
1. Counting the degrees:
- Each vertex in the graph has a degree, which represents the number of edges incident to that vertex.
- The sum of the degrees of all the vertices can be denoted as Σ(deg(v)), where deg(v) represents the degree of vertex v.
2. Relationship between degrees and edges:
- In an undirected graph, each edge connects two vertices.
- Therefore, each edge contributes 1 to the degree of each of its incident vertices.
- Since there are n edges in the graph, each edge contributes 2 to the sum of the degrees.
3. Sum of the degrees:
- The sum of the degrees can be calculated as follows:
Σ(deg(v)) = 2n
- This is because each edge contributes 2 to the sum of the degrees, and there are n edges in total.
Now, we need to find the sum of the degrees for an undirected graph with m vertices and n edges.
4. Substituting the values:
- Since the graph has m vertices, the sum of the degrees can be expressed as Σ(deg(v)) = 2n = 2m.
- This means that the sum of the degrees of all the vertices is equal to 2m.
Therefore, the correct answer is option 'C': 2m.
Let's break down the problem step by step:
1. Counting the degrees:
- Each vertex in the graph has a degree, which represents the number of edges incident to that vertex.
- The sum of the degrees of all the vertices can be denoted as Σ(deg(v)), where deg(v) represents the degree of vertex v.
2. Relationship between degrees and edges:
- In an undirected graph, each edge connects two vertices.
- Therefore, each edge contributes 1 to the degree of each of its incident vertices.
- Since there are n edges in the graph, each edge contributes 2 to the sum of the degrees.
3. Sum of the degrees:
- The sum of the degrees can be calculated as follows:
Σ(deg(v)) = 2n
- This is because each edge contributes 2 to the sum of the degrees, and there are n edges in total.
Now, we need to find the sum of the degrees for an undirected graph with m vertices and n edges.
4. Substituting the values:
- Since the graph has m vertices, the sum of the degrees can be expressed as Σ(deg(v)) = 2n = 2m.
- This means that the sum of the degrees of all the vertices is equal to 2m.
Therefore, the correct answer is option 'C': 2m.
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