Let’s delve into each part of this problem.

Part (a): Number of paths of length 2 in $G$

1. Identify paths of length 2 involving $c$: Since $c$ is connected to every vertex in $A$, any path of length 2 can be in the form $a_i \to c \to a_j$ where $a_i, a_j \in A$ and $a_i \neq a_j$. For each vertex $a_i$ in $A$, there are $n - 1$ choices for $a_j$, giving $n \times (n - 1)$ paths of this form.

2. Paths within the cycle $A$: In the cycle induced by $A$, every pair of consecutive vertices can form a 2-step path by going from $a_i$ to $a_{i+1}$ and then to $a_{i+2}$ (mod $n$ to keep the cycle intact). Since there are $n$ vertices in $A$, there are $n$ such paths in the cycle.

Therefore, the total number of paths of length 2 in $G$ is: $n \times (n - 1) + n = n(n - 1) + n = n^2$

Part (b): Number of cycles in $G$

1. Cycle within $A$: The subgraph induced by $A$ forms a cycle of length $n$, so there is exactly one cycle of length $n$ here.

2. Cycles involving $c$: For cycles involving $c$, we can select any two distinct vertices $a_i, a_j$ in $A$ and form a 3-cycle $c \to a_i \to a_j \to c$. There are $\binom{n}{2} = \frac{n(n - 1)}{2}$ ways to choose such pairs $(a_i, a_j)$, so there are $\frac{n(n - 1)}{2}$ cycles of length 3 that involve $c$.

Thus, the total number of cycles in $G$ is: $1 + \frac{n(n - 1)}{2}$

Would you like to discuss the reasoning further or explore related concepts? Here are some related questions:

1. What would change if $c$ were connected to only some vertices in $A$ instead of all?
2. How would the graph properties change if $A$ formed a complete graph instead of a cycle?
3. What are the implications of adding more vertices to $A$ while keeping $c$ connected to each vertex?
4. How would the number of paths of different lengths vary with changes in $n$?
5. Could we classify this graph based on standard graph theory types (e.g., bipartite, planar)?

Tip: When analyzing paths and cycles in graphs, consider breaking down the graph structure into simpler components, such as cycles and connections, to streamline counting paths and cycles.