2022-083

A Cayley tree is a tree in which each non-leaf vertex has a constant degree \( k \), and there is a central vertex with distance \( P \) to all leaves. Here is a visual example of this type of graph, for \( k=3 \) and \( P=6 \):

Consider the following statements about Cayley trees:

1. A Cayley Tree is a connected graph with no cycles
2. Cayley Trees with \( k \ge 3 \) form an infinite graph family of networks with the small-word property
3. In the \( G(N, 0.3) \) model, the probability of a graph being a Cayley tree is zero for infinitely many values of \( N \)
4. The clustering coefficient of any node in a Cayley Tree is zero

Now select the alternative listing exactly the true statements:

1. All statements are correct
2. I and IV
3. II and III
4. I and II
5. None of the above

Original idea by: Victória Pedrazzoli