Let’s analyze the structure of the graph $G = (V, E)$ given in the problem and answer each question systematically.

1. Graph Structure:
   • Vertices: The set of vertices $V = A \cup \{c\}$ has $n+1$ vertices, where $A = \{a_1, \ldots, a_n\}$ is a set of $n$ vertices and $c \notin A$ is an additional vertex.
   • Edges:
     • The vertices in $A$ form a cycle, which means each vertex $a_i$ is connected to $a_{i+1}$, and the last vertex $a_n$ is connected back to $a_1$.
     • The vertex $c$ is connected to every vertex in $A$, making it a central point in the graph that connects to all other vertices in $A$.
     • No other edges exist in $G$.

Given this structure, let’s answer parts (a) and (b) in detail.

Part (a): Number of Paths of Length 2

A path of length 2 is a sequence of three vertices $(v_1, v_2, v_3)$ where $(v_1, v_2) \in E$ and $(v_2, v_3) \in E$. We will calculate the number of such paths by considering two cases based on the position of $c$.

Case 1: $c$ as the Middle Point

If $c$ is the middle vertex of the path, we have a path $(a_i, c, a_j)$ where both $a_i$ and $a_j$ are vertices in $A$, and they are distinct since no self-loops are allowed. For each $a_i$ in $A$, we can pair it with any other $a_j$ in $A$, giving us $n(n - 1)$ paths of this type.

Case 2: $c$ as the End Point

If $c$ is an endpoint of the path, we have two possibilities:

1. The path is $(c, a_i, a_{i+1})$ for each edge $(a_i, a_{i+1})$ in the cycle of $A$.
2. The path is $(a_i, a_{i+1}, c)$ for each edge $(a_i, a_{i+1})$ in the cycle of $A$.

Since there are $n$ edges in the cycle of $A$, we have $2n$ paths of this type (one for each direction).

Total Number of Paths of Length 2

Adding up the paths from both cases, we get: $\text{Total paths of length 2} = n(n - 1) + 2n = n^2$

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

A cycle is a closed path where the starting and ending vertices are the same, and each edge is used exactly once.

Case 1: Cycles in $A$

Since $A$ forms a simple cycle by itself, there is exactly one cycle involving only the vertices in $A$: $a_1 \to a_2 \to \ldots \to a_n \to a_1$

Case 2: Cycles Including $c$ as a Vertex

To form a cycle that includes $c$, we can pick any two distinct vertices $a_i$ and $a_j$ from $A$ and construct a cycle $(a_i, c, a_j, a_{i-1}, \ldots, a_j)$.