The problem in the image asks to determine whether the given graphs $G$ and $H$ are isomorphic, and to justify the answer with appropriate reasons.

Step 1: Understanding Graph Isomorphism

Two graphs are isomorphic if there is a one-to-one correspondence between their vertices and edges, such that the adjacency relationships between vertices are preserved. In simpler terms:

• The number of vertices in both graphs must be the same.
• The number of edges must be the same.
• The degree (number of edges) of corresponding vertices should match.
• The structure (connections) should be preserved.

Step 2: Analyzing the Graphs $G$ and $H$

Graph $G$:

Vertices: $a, b, c, d, e, f, g, h$

Edges (pairs of connected vertices):

• $a$--$b$, $b$--$c$, $c$--$d$, $d$--$e$, $e$--$f$, $f$--$g$, $g$--$h$, $h$--$a$

Graph $H$:

Vertices: $w, x, y, z, u, v, s, t$

Edges (pairs of connected vertices):

• $w$--$x$, $x$--$y$, $y$--$z$, $z$--$u$, $u$--$v$, $v$--$s$, $s$--$t$, $t$--$w$

Step 3: Comparison

• Number of vertices: Both $G$ and $H$ have 8 vertices.
• Number of edges: Both graphs also seem to have 8 edges.
• Degree of vertices: Every vertex in both graphs has degree 2 (i.e., each vertex is connected to exactly two other vertices).
• Adjacency preservation: The structure of both graphs seems to follow the same pattern—a cycle connecting all vertices. The labeling differs, but the overall connection structure appears identical.

Conclusion:

Since the graphs $G$ and $H$ have the same number of vertices and edges, the same vertex degrees, and a similar cyclic structure, they are isomorphic.

Let me know if you'd like further details or have any questions!

Here are 5 related questions to deepen understanding:

1. What are the formal steps to prove two graphs are isomorphic?
2. Can two graphs with the same number of vertices and edges but different structures be isomorphic?
3. What is the role of vertex degrees in determining graph isomorphism?
4. How can automorphisms help in analyzing graph symmetries?
5. Are there efficient algorithms for checking graph isomorphisms in large graphs?

Tip: When checking if two graphs are isomorphic, always start by comparing their basic properties—vertices, edges, and vertex degrees—before diving into more complex structural analysis.