Five people are all connected by e-mail. Whenever one of them hears a juicy piece of

Chapter 3, Problem 3.7.71

(choose chapter or problem)

Five people are all connected by e-mail. Whenever one of them hears a juicy piece of gossip, he or she passes it along by e-mailing it to someone else in the group according to Table 3.6. (a) Draw the digraph that models this gossip network and find its adjacency matrix A.(b) Define a step as the time it takes a person to e-maileveryone on his or her list. (Thus, in one step, gossipgets from Ann to both Carla and Ehaz.) If Berthears a rumor, how many steps will it take foreveryone else to hear the rumor? What matrixcalculation reveals this?(c) If Ann hears a rumor, how many steps will it takefor everyone else to hear the rumor? What matrixcalculation reveals this?(d) In general, if A is the adjacency matrix of adigraph, how can we tell if vertex i is connected tovertex j by a path (of some length)?[The gossip network in this exercise is reminiscentof the notion of six degrees of separation (found inthe play and film by that name), which suggests thatany two people are connected by a path of acquaintanceswhose length is at most 6. The game SixDegrees of Kevin Bacon more frivolously assertsthat all actors are connected to the actor Kevin Baconin such a way.]