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(a) Give an example of a graph in which every node is pivotal for at least one pair of nodes. Explain your answer.
(a) Give an example of a graph in which every node is pivotal for at least one pair of nodes. Explain your answer.(b) Give an example of a graph in which every node is pivotal for at least two different pairs of nodes. Explain your answer.(c) Give an example of a graph having at least four nodes in which there is a single node X that is pivotal for every pair of nodes (not counting pairs that includeX). Explain your answer.(d) Give an example (together with an explanation) of a graph in which more than half of all nodes are gatekeepers.(e) Give an example (together with an explanation) of a graph in which there are no gatekeepers, but in which every node is a local gatekeeper.(f) Describe an example of a graph where the diameter is more than three times as large as the average distance.(g) Describe how you could extend your construction to produce graphs in which the diameter exceeds the average distance by as large a factor as you'd like. (That is,for every number c, can you produce a graph in which the diameter is more than c times as large as the average distance?)
