To prove that two graphs $G_1$ and $G_2$ are isomorphic, we must show there exists a one-to-one correspondence (bijection) between their vertices and edges such that adjacency is preserved. This means that there should be a mapping between the vertices of $G_1$ to those of $G_2$, and for any two adjacent vertices in $G_1$, their corresponding vertices in $G_2$ should also be adjacent.

Let's go through the general steps for proving graph isomorphism:

1. Check the number of vertices and edges

• The graphs must have the same number of vertices and edges. If this condition fails, they cannot be isomorphic.

2. Check the degrees of vertices

• The degree of each vertex (the number of edges incident to it) in $G_1$ must correspond to the degree of a vertex in $G_2$. A one-to-one correspondence must be established between the vertices of $G_1$ and $G_2$ based on their degrees.

3. Find a vertex mapping

• Map vertices of $G_1$ to $G_2$ based on the degrees, ensuring that the number of neighbors for each vertex in the first graph matches the number of neighbors in the second graph.

4. Check adjacency preservation

• For each pair of adjacent vertices in $G_1$, their corresponding vertices in $G_2$ must also be adjacent. This ensures that the edges are preserved in the mapping.

5. Verify all edges

• After mapping all vertices, check if the adjacency relationships (edges) are preserved under the vertex mapping. If every edge of $G_1$ corresponds to an edge in $G_2$ in the same manner, the graphs are isomorphic.

Example of Graph Isomorphism:

Consider two graphs:

• $G_1$ has vertices $\{ A, B, C, D \}$ with edges $\{ AB, AC, BD, CD \}$.
• $G_2$ has vertices $\{ W, X, Y, Z \}$ with edges $\{ WX, WY, XZ, YZ \}$.

Step 1: Check the number of vertices and edges

• Both graphs have 4 vertices and 4 edges. This condition is satisfied.

Step 2: Check vertex degrees

• For $G_1$:

  • $A$ has degree 2 (adjacent to $B$ and $C$).
  • $B$ has degree 2 (adjacent to $A$ and $D$).
  • $C$ has degree 2 (adjacent to $A$ and $D$).
  • $D$ has degree 2 (adjacent to $B$ and $C$).

• For $G_2$:

  • $W$ has degree 2 (adjacent to $X$ and $Y$).
  • $X$ has degree 2 (adjacent to $W$ and $Z$).
  • $Y$ has degree 2 (adjacent to $W$ and $Z$).
  • $Z$ has degree 2 (adjacent to $X$ and $Y$).

Both graphs have the same degree distribution, so this condition is satisfied.

Step 3: Find a vertex mapping

• Based on degrees, let's try this mapping:
  • $A$ ↔ $W$
  • $B$ ↔ $X$
  • $C$ ↔ $Y$
  • $D$ ↔ $Z$

Step 4: Check adjacency preservation

• For $G_1$:
  • $AB$ corresponds to $WX$
  • $AC$ corresponds to $WY$
  • $BD$ corresponds to $XZ$
  • $CD$ corresponds to $YZ$

This mapping preserves all adjacencies, so the graphs are isomorphic.

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Would you like to see another example or explore specific details about graph isomorphism? Here are 5 related questions:

1. What are the necessary conditions for two graphs to be isomorphic?
2. How do we efficiently find a graph isomorphism using algorithms?
3. How do we prove that two graphs are non-isomorphic?
4. Can two non-connected graphs be isomorphic to each other?
5. What is the role of adjacency matrices in proving graph isomorphism?

Tip: If you're working with large graphs, consider using software tools like Graph isomorphism algorithms or libraries (e.g., NetworkX in Python) to check isomorphisms automatically.