Let's break down the construction and analyze the questions:

Part (a): Number of Paths of Length 2 in $G$

1. Understanding Paths of Length 2: A path of length 2 is a sequence of three vertices $(v_1, v_2, v_3)$ where $v_1$ is connected to $v_2$ and $v_2$ is connected to $v_3$. Thus, we need to count all such sequences in $G$.

2. Types of Paths:

   • Case 1: Paths where $c$ is the middle vertex. For a path of length 2 with $c$ as the middle vertex, we have:

     • Start at a vertex $a_i$ in $A$, go to $c$, then go to another vertex $a_j$ in $A$.
     • For each pair $(a_i, a_j)$ in $A$, there is such a path. Since there are $n$ choices for $a_i$ and $n-1$ remaining choices for $a_j$, this gives a total of $n(n - 1)$ paths of this type.

   • Case 2: Paths within the cycle induced by $A$ (i.e., not involving $c$). Each vertex in $A$ has two neighbors in the cycle, so for each $a_i$, there are two paths of length 2 that pass through it. This yields $2n$ paths in this case (as there are $n$ vertices in $A$, and each has two such paths).

3. Total Number of Paths of Length 2: $n(n - 1) + 2n = n^2$

Thus, the total number of paths of length 2 in $G$ is $\boxed{n^2}$.

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

1. Understanding Cycles in $G$: A cycle in a graph is a closed path where the starting and ending vertices are the same, and no other vertices are repeated. In this graph, cycles can form either within the set $A$ or by including the vertex $c$.

2. Types of Cycles:

   • Case 1: Cycles within $A$. Since $A$ induces a cycle, there is exactly one $n$-cycle among the vertices in $A$, namely $a_1 \to a_2 \to \dots \to a_n \to a_1$.

   • Case 2: Cycles involving $c$. For each pair $(a_i, a_j)$ of distinct vertices in $A$, we can form a 3-cycle $(a_i, c, a_j, a_i)$. The number of such pairs $(a_i, a_j)$ is $\binom{n}{2} = \frac{n(n - 1)}{2}$, so there are $\frac{n(n - 1)}{2}$ 3-cycles of this type.

3. Total Number of Cycles: $1 + \frac{n(n - 1)}{2}$

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

Discussion of $c$ as Middle Point and End Point

• When $c$ is the middle point: This scenario occurs in paths of length 2 where the path starts and ends at vertices in $A$ with $c$ in between (i.e., paths of the form $(a_i, c, a_j)$ for distinct $a_i, a_j$ in $A$). These paths contribute to the count of paths of length 2 discussed in Part (a).

• When $c$ is the end point: This scenario doesn’t contribute to the length 2 paths as defined in the problem (since they require paths with two edges and therefore an intermediate vertex). However, it does contribute to the construction of the graph $G$ by connecting each vertex in $A$ to $c$, enabling the formation of cycles and certain paths.

Would you like further clarification on any part?

Here are five related questions:

1. What would change if we allowed self-loops in the graph $G$?
2. How would the total path and cycle counts differ if $G$ was directed?
3. Can you find a general formula for paths of length $k$ in this graph?
4. What other properties of $G$ can we infer from this structure (e.g., connectivity, bipartiteness)?
5. How does the structure of $G$ change if we add more vertices connected only to $c$?

Tip: When analyzing paths in a graph, breaking them down by cases (such as which vertex serves as the middle) can simplify calculations.