Let A = {a1,..., an} be a set with n ≥ 3 elements and let c $ A.
Construct a graph G = (V, E) as follows. The set of vertices is V =
AU{c} and the set of edges E is defined by
• The sub-graph of G induced by A is a cycle.
• There is an edge between c and every vertex in A.
• There are no other edges except those specified above.
(a) Express the number of paths of length 2 in G in terms of n. Justify your answer.
(b) Express the number of cycles in G in terms of n. Justify your answer.

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.

Let A = {a1,..., an} be a set with n ≥ 3 elements and let c ∈ A. Construct a graph G = (V, E) as follows. The set of vertices is V = A ∪ {c} and the set of edges E is defined by: • The sub-graph of G induced by A is a cycle. • There is an edge between c and every vertex in A. • There are no other edges except those specified above. (a) Express the number of paths of length 2 in G in terms of n. Justify your answer. (b) Express the number of cycles in G in terms of n. Justify your answer.

Explore a unique graph theory problem involving paths and cycles. Learn how to calculate the number of paths of length 2 and the number of cycles in a graph constructed with a set A and additional vertex c. This solution covers essential graph theory and combinatorics concepts suitable for advanced high school and university students.

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