1. **Stating the problem:** We need to understand the theorem of isomorphism for graphs, which states that if two graphs $G$ and $G'$ are isomorphic ($G \cong G'$), then they share several properties. 2. **Theorem statement:** If $G \cong G'$, then: - They have the same number of vertices. - They have the same number of edges. - They have the same degree sequence. - They have the same number of cycles of different lengths. 3. **Explanation:** Two graphs are isomorphic if there is a one-to-one correspondence between their vertex sets that preserves adjacency. 4. **Example 1:** Consider graph $G$ with vertices $\{A,B,C,D\}$ and edges $\{AB, BC, CD, DA\}$ forming a square. - Number of vertices: 4 - Number of edges: 4 - Degree sequence: $(2,2,2,2)$ - Cycles: One cycle of length 4 5. **Example 2:** Consider graph $G'$ with vertices $\{W,X,Y,Z\}$ and edges $\{WX, XY, YZ, ZW\}$ also forming a square. - Number of vertices: 4 - Number of edges: 4 - Degree sequence: $(2,2,2,2)$ - Cycles: One cycle of length 4 6. **Verification:** Since $G$ and $G'$ have the same number of vertices, edges, degree sequence, and cycles, they are isomorphic. 7. **Conclusion:** The theorem holds as these properties are invariant under graph isomorphism. Final answer: Two graphs $G$ and $G'$ are isomorphic if and only if they have the same number of vertices, edges, degree sequence, and cycles of different lengths.