To determine if each pair of graphs in the images are isomorphic, let's analyze the structure of each graph in pairs, noting key graph properties such as the number of vertices, edges, and vertex degree sequences.

Part (i): Graphs $u$ and $v$

1. Vertices: Both graphs $u$ and $v$ have 8 vertices.
2. Edges: Both graphs have the same number of edges.
3. Vertex Degrees:
   • In the left graph $u$, each vertex has degree 3. This suggests it may represent a simple cycle with some additional connections.
   • In the right graph $v$, each vertex has degree 7, indicating it is a complete graph on 8 vertices, $K_8$, where every vertex is connected to every other vertex.

Since the degree sequences differ (all vertices in $u$ have degree 3, while all vertices in $v$ have degree 7), these two graphs are not isomorphic. A necessary condition for graph isomorphism is having the same degree sequence, which is violated here.

Part (ii): Graphs $e$ and $f$

1. Vertices: Both graphs have 5 vertices.
2. Edges:
   • The left graph $e$ has 7 edges.
   • The right graph $f$ also has 7 edges.
3. Vertex Degrees:
   • For graph $e$, the degree sequence appears to be: 3, 3, 2, 2, 2.
   • For graph $f$, the degree sequence appears to be: 3, 3, 2, 2, 2.

The degree sequences match, suggesting they could be isomorphic. Now, let’s identify a possible mapping (isomorphism) between the vertices of $e$ and $f$ that preserves adjacency:

• Let $v_1 \leftrightarrow w_1$, $v_2 \leftrightarrow w_2$, $v_3 \leftrightarrow w_3$, $v_4 \leftrightarrow w_4$, $v_5 \leftrightarrow w_5$.
• Check that the edges correspond under this mapping:
  • The edges $\{v_1, v_2\}, \{v_1, v_5\}, \{v_1, v_3\}$ in $e$ correspond to $\{w_1, w_2\}, \{w_1, w_5\}, \{w_1, w_3\}$ in $f$.
  • Similarly, the other edges in $e$ correspond to those in $f$ under this mapping.

Since we can find a one-to-one correspondence that preserves edges, graphs $e$ and $f$ are isomorphic.

Summary

1. Graphs $u$ and $v$ are not isomorphic due to differing vertex degrees.
2. Graphs $e$ and $f$ are isomorphic, with an isomorphism that matches vertices based on the described mapping.

Would you like a deeper explanation or have any further questions?

Further Questions:

1. What are the general criteria for two graphs to be isomorphic?
2. How can we systematically check if two graphs are isomorphic using adjacency matrices?
3. What other properties, besides vertex degree, could help determine graph isomorphism?
4. How does the structure of a complete graph differ from other types of graphs?
5. Why is vertex degree a key property when comparing graphs for isomorphism?

Tip:

When checking for graph isomorphisms, start with comparing basic properties like the number of vertices, edges, and degree sequences before attempting to map vertices.