This question asks to determine whether the pairs of graphs in parts (a) and (b) are isomorphic. Isomorphic graphs are those that have the same number of vertices, edges, and a one-to-one correspondence between their vertices such that adjacency is preserved.

Solution:

Part (a): Two graphs

1. Count the vertices and edges:

   • Both graphs have 7 vertices and 12 edges.

2. Degrees of vertices:

   • For Graph 1: Vertex degrees are $3, 3, 3, 3, 3, 3, 3$ (all vertices have degree 3).
   • For Graph 2: Vertex degrees are also $3, 3, 3, 3, 3, 3, 3$.

3. Graph structure:

   • Both graphs appear to be representations of the cycle graph $C_7$ with additional edges connecting non-adjacent vertices (essentially creating a generalized wheel graph). These two graphs are isomorphic because their degree distributions and edge connections can be mapped while preserving adjacency.

Part (b): Two graphs

1. Count the vertices and edges:

   • Both graphs have 8 vertices and 12 edges.

2. Degrees of vertices:

   • For Graph 1: Degrees are $3, 3, 3, 3, 3, 3, 3, 3$.
   • For Graph 2: Degrees are $3, 3, 3, 3, 3, 3, 3, 3$.

3. Graph structure:

   • Both graphs form a cube-like structure (a 3D representation of a cube projected onto 2D). This means they are also isomorphic, as the same vertices and edges can be matched between the graphs, maintaining adjacency.

Conclusion:

• Part (a): The two graphs are isomorphic.
• Part (b): The two graphs are isomorphic.

Would you like a more detailed explanation of the isomorphism mapping for these graphs?

5 Related Questions:

1. What are some practical methods for checking graph isomorphism?
2. How can adjacency matrices be used to verify isomorphic graphs?
3. What is the degree sequence of a graph, and how is it useful for isomorphism?
4. What are the properties of cycle graphs and cube graphs?
5. Can two graphs with different edge counts ever be isomorphic?

Tip:

When checking for isomorphism, focus on degree distributions, number of vertices/edges, and graph invariants like cycles, connectivity, and adjacency relations!