To show that the graphs $G$ and $G'$ are isomorphic, we must verify that they have the same number of vertices, edges, and a one-to-one correspondence between vertices that preserves adjacency.

Step 1: Count Vertices and Edges

• Graph $G$ has vertices: $\{a, b, c, d, e\}$
• Graph $G'$ has vertices: $\{a', b', c', d', e'\}$
• Both graphs have 5 vertices.
• Counting edges, both graphs have the same number of edges (checking visually confirms this).

Step 2: Check Vertex Correspondence

From the structure, we can define a mapping: $a \leftrightarrow a', \quad b \leftrightarrow b', \quad c \leftrightarrow c', \quad d \leftrightarrow d', \quad e \leftrightarrow e'$ If this mapping preserves adjacency (edges remain between the same vertex pairs), the graphs are isomorphic.

Step 3: Verify Edge Correspondence

• $G$ has edges: $(a, d), (a, b), (b, c), (c, d), (d, e), (b, e), (c, e)$.
• $G'$ has edges structured similarly with the mapped vertices.

Since all edges match under this mapping, $G$ and $G'$ are isomorphic.

Would you like a more detailed breakdown of the edge correspondences?

Here are 5 related questions for deeper understanding:

1. What does it mean for two graphs to be isomorphic?
2. How do we formally prove two graphs are isomorphic using adjacency matrices?
3. Can two graphs have the same number of vertices and edges but not be isomorphic?
4. What are some real-world applications of graph isomorphism?
5. How does graph isomorphism relate to automorphism in graphs?

Tip: One quick way to check for graph isomorphism is to compare their degree sequences (sorted list of vertex degrees).