1.	Examine whether the following pair of graphs are isomorphic. If not isomorphic, give the reasons.
(AU A/M 2011)
 


To determine if two graphs are isomorphic, we need to verify if there is a one-to-one correspondence (a bijection) between their vertex sets such that the edges are preserved. Two graphs are isomorphic if they have the same number of vertices, the same number of edges, and the same degree sequence for each vertex. Additionally, there must be a way to map vertices in one graph to vertices in the other, such that adjacent vertices remain adjacent in the mapping.

Without seeing the pair of graphs you are referring to, I can't provide a specific solution. However, here’s a general outline for how you would check for isomorphism:

### Steps to check if two graphs are isomorphic:

1. **Check number of vertices and edges:**
   - If the graphs have a different number of vertices or edges, they cannot be isomorphic.

2. **Check degree sequences:**
   - The degree sequence (list of vertex degrees) must be the same for both graphs. If the degree sequences differ, the graphs are not isomorphic.

3. **Check edge connections:**
   - If the degree sequences are identical, try to map vertices of the first graph to vertices of the second graph, preserving the edges. This means that if two vertices are adjacent in the first graph, their mapped counterparts must be adjacent in the second graph.

4. **Automorphisms and adjacency matrix:**
   - Another way to check isomorphism is by constructing the adjacency matrix for both graphs and seeing if one can be transformed into the other by rearranging rows and columns (i.e., through permutations that preserve the adjacency).

5. **Check for distinct substructures:**
   - If one graph has a substructure (like a cycle, clique, or bipartite form) that the other does not have, they are not isomorphic.

If you have the specific graphs, feel free to upload them and I can help you analyze their isomorphism in detail.

Would you like to know more about any of these steps or go deeper into specific examples?

### Related Questions:
1. How can you use adjacency matrices to test for graph isomorphism?
2. What are common graph invariants used in detecting non-isomorphic graphs?
3. How does the degree sequence help in checking graph isomorphism?
4. Can isomorphic graphs have different representations?
5. What are automorphisms in graph theory and how do they relate to isomorphisms?

**Tip:** When comparing graphs for isomorphism, start with simple checks like vertex and edge counts before diving into more complex properties like adjacency or automorphisms.

Examine whether the following pair of graphs are isomorphic. If not isomorphic, give the reasons. (AU A/M 2011)

Explore how to determine if two graphs are isomorphic using degree sequences, edge connections, and adjacency matrices. This AU A/M 2011 problem helps students understand the core concepts of graph theory and graph isomorphism. Suitable for college-level students.

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