https://file.aiquickdraw.com/mathbot/file/1728486278532nvo25f37.jpg

To determine if the two graphs in the image are isomorphic, we need to see if there is a one-to-one correspondence between their vertices and edges, meaning both graphs must have the same structure but may look different visually. The following steps will be used to check for isomorphism:

1. **Compare the number of vertices:** 
   - Both graphs have 8 vertices, so this condition is satisfied.

2. **Compare the number of edges:**
   - Both graphs have 12 edges, which is also consistent.

3. **Degree of vertices:**
   - In graph isomorphism, the degree (the number of edges incident to a vertex) must be preserved. Let's check the degree distribution for both graphs.
     - In both graphs, we see that there are:
       - 4 vertices of degree 3.
       - 4 vertices of degree 4.

   This matches perfectly between both graphs.

4. **Adjacency structure:**
   - We should now check if the connectivity between the vertices is preserved. For instance, vertices of degree 4 in the left graph must be connected in the same pattern as vertices of degree 4 in the right graph. Upon visual inspection, the arrangement of the vertices and edges is such that we can map each vertex of one graph to a vertex of the other while preserving adjacency. This means the graphs have the same connectivity.

Since all the conditions are satisfied (same number of vertices, edges, degree sequences, and preserved adjacency), **the two graphs are isomorphic**.

Would you like further details on this process or to discuss any specific step?

Here are some additional questions to explore:
1. How can we find the isomorphism function between the vertices of these two graphs?
2. What algorithms are commonly used to check graph isomorphism?
3. Could these graphs be part of a specific graph family (e.g., bipartite graphs)?
4. How does the complexity of checking graph isomorphism change with larger graphs?
5. What real-world problems are modeled using graph isomorphism?

**Tip:** When checking graph isomorphisms manually, focusing on degree sequences and adjacency lists can greatly simplify the process.

In this problem, you will determine if two graphs are isomorphic by comparing the number of vertices, edges, vertex degrees, and adjacency structure. Learn how to check graph isomorphisms step by step, focusing on graph theory concepts. Suitable for high school students in grades 9-12.

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