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.
a. Draw the digraph that models this gossip network and find its adjacency matrix A.
Sender Recipients
Ann ........................... Carla, Ehaz
Bert ........................... Carla, Dana
Carla .................................. Ehaz
Dana ........................... Ann, Carla
Ehaz .................................... Bert
b. Define a step as the time it takes a person to e-mail everyone on his or her list. Thus, in one step gossip gets from Ann to both Carla and Ehaz. If Bert hears a rumor, how many steps will it take for everyone else to hear the rumor? What matrix calculation reveals this?
c. If Ann hears a rumor, how many steps will it take for everyone else to hear the rumor? What matrix calculation reveals this?
d. In general, if A is the adjacency matrix of a digraph, how can we tell if vertex I is connected to vertex j by a path (of some length)?
[The gossip network in this exercise is reminiscent of the notion of six degrees of separation (found in the play and film by that name), which suggests that any two people are connected by a path of acquaintances whose length is at most 6. The game Six Degrees of Kevin Bacon more frivolously asserts that all actors are connected to the actor Kevin Bacon in such a way.