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Check whether these two graphs are isomorphic or not. Consider Vertex mapping

To check whether two graphs are isomorphic, we need to verify if there exists a one-to-one correspondence (bijection) between the vertex sets of the two graphs such that the adjacency relations are preserved. This means that two vertices that are adjacent in one graph must map to two vertices that are adjacent in the other graph.

### Graph G (left):
- Vertices: \( \{a, b, c, d, e, f\} \)
- Edges: \( (a,b), (a,f), (b,c), (c,d), (d,e), (e,f) \)

### Graph H (right):
- Vertices: \( \{u, v, w, x, y, z\} \)
- Edges: \( (u,v), (u,z), (v,w), (w,x), (x,y), (y,z) \)

Both graphs have 6 vertices and 6 edges, so they could potentially be isomorphic. Let's attempt to find a vertex mapping that preserves adjacency.

### Step-by-Step Vertex Mapping:

1. Both graphs have a cycle of length 6, which suggests that corresponding vertices in the cycle can be mapped to each other.
   
2. Let's start by mapping:
   - \( a \mapsto u \) (since both are connected to two vertices)
   - \( b \mapsto v \)
   - \( f \mapsto z \)
   - \( e \mapsto y \)
   - \( d \mapsto x \)
   - \( c \mapsto w \)

### Adjacency Check:
- \( (a,b) \mapsto (u,v) \): Correct
- \( (a,f) \mapsto (u,z) \): Correct
- \( (b,c) \mapsto (v,w) \): Correct
- \( (c,d) \mapsto (w,x) \): Correct
- \( (d,e) \mapsto (x,y) \): Correct
- \( (e,f) \mapsto (y,z) \): Correct

Thus, the adjacency relations are preserved under this mapping. Therefore, **the two graphs are isomorphic**.

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Would you like more details or have any other questions?

Here are 5 related questions to explore:
1. What are some other methods for checking graph isomorphism?
2. Can two graphs with different degrees of vertices be isomorphic?
3. What is the time complexity of checking graph isomorphism for large graphs?
4. How do isomorphic graphs relate to automorphisms?
5. Can isomorphism be used to solve real-world problems?

**Tip:** In practice, adjacency matrices or characteristic polynomials can be used to verify graph isomorphism quickly.

Check whether these two graphs are isomorphic or not.

This problem involves checking whether two given graphs are isomorphic by examining vertex mapping and adjacency relations. The solution discusses key concepts from graph theory, such as graph isomorphism and adjacency, providing a step-by-step mapping of vertices to confirm the graphs' equivalence.

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