A Cayley tree is a symmetric regular tree in which each node is connected to the...

 

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A Cayley tree is a symmetric regular tree in which each node is connected to the same number k of others, until we get out to the leaves, like this: (We have k = 3 in this picture.) Show that the number of nodes reachable in d steps from the central node is k(k 1)d-1 for d 1. Hence find an expression for the diameter of the network in terms of k and the number of nodes n. A Cayley tree is a symmetric regular tree in which each node is connected to the same number k of others, until we get out to the leaves, like this: (We have k = 3 in this picture.) Show that the number of nodes reachable in d steps from the central node is k(k 1)d-1 for d 1. Hence find an expression for the diameter of the network in terms of k and the number of nodes n.