handshaking theorem show that the number of edges of the graph is a multiple of k, if all the vertices of an undirectional graph are each of degree k

The **Handshaking Theorem** states that in any undirected graph, the sum of the degrees of all vertices is equal to twice the number of edges. Formally:

\[
\sum_{v \in V} \text{deg}(v) = 2|E|
\]

Where:
- \(\text{deg}(v)\) is the degree of vertex \(v\),
- \(|E|\) is the number of edges in the graph.

Now, let's prove that the number of edges \(|E|\) is a multiple of \(k\) if all vertices in the graph have the same degree \(k\):

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### Proof:
1. **Given**: All vertices in the graph have degree \(k\), and let the number of vertices be \(n\).
   - Total degree of all vertices: \( \sum_{v \in V} \text{deg}(v) = n \cdot k \).

2. **By the Handshaking Theorem**:
   \[
   \sum_{v \in V} \text{deg}(v) = 2|E|
   \]
   Substituting the total degree:
   \[
   n \cdot k = 2|E|
   \]

3. Rearranging the equation for \(|E|\):
   \[
   |E| = \frac{n \cdot k}{2}
   \]

4. **Analysis**: For \(|E|\) to be an integer, \(n \cdot k\) must be even. Since \(k\) is the degree of each vertex:
   - If \(k\) is even, \(n \cdot k\) is always even (regardless of \(n\)).
   - If \(k\) is odd, \(n\) must be even to ensure \(n \cdot k\) is even (this is always true for undirected graphs with all vertices of the same degree, as an odd degree for all vertices implies an even number of vertices).

5. **Conclusion**: \(|E|\) is an integer and a multiple of \(k\), as shown in the equation \(|E| = \frac{n \cdot k}{2}\), where \(n \cdot k\) is divisible by \(2k\).

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This proof confirms that if all vertices of an undirected graph have the same degree \(k\), the number of edges \(|E|\) is a multiple of \(k\).

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Would you like further examples or applications of this theorem? Here are 5 related questions for exploration:

1. What happens if the graph is directed? Does the Handshaking Theorem still apply?
2. Can you provide an example graph where all vertices have degree 3 and calculate \(|E|\)?
3. How does the Handshaking Theorem relate to bipartite graphs?
4. What is the significance of the Handshaking Theorem in social network analysis?
5. Can this theorem be used to detect graph anomalies (e.g., errors in degree assignments)?

**Tip**: The Handshaking Theorem is useful for verifying the structure of graphs—use it to quickly check for inconsistencies in degree distributions!

Handshaking theorem show that the number of edges of the graph is a multiple of k, if all the vertices of an undirectional graph are each of degree k.

Explore the Handshaking Theorem and its proof that the number of edges in an undirected graph is a multiple of k when all vertices have the same degree k. Learn key graph theory concepts, formulas, and theorems for understanding this property.

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