Application

A wired network is a set of computers and ethernet connections.

Each ethernet connection is between exactly two computers in the network. If

there is an ethernet connection between computers $C$ and $D,$ then we can say that

$C$ connects to $D$ and $D$ connects to $C.$

A pair of computers can have at most one ethernet connection between them.

A data line between distinct computers $C,D$ is a sequence of connections that

starts at $C$ and concludes at $D.$

A connected wired network is a wired network where every pair of computers has

a data line between them.

Assume computers can be constructed with any number of ethernet ports to

support being participants in any number of ethernet connections.

To illustrate, $C,D$ could have the sequence $((C,C_1),(C_1,C_2),$dots,$(C_n,D))$ where

$C$ connects to $C_1,$ then $C_1$ connects to $C_2,$ and so on to $C_n$ which connects to $D.$

For example, consider the set of computers $\{A,B,C,D\}$ where we have the ethernet

connections set $\{(A,B),(B,C),(A,D)\}.$ This is a connected wired network as every

computer can reach every other computer via a data line. This can be seen below.

$($a$)$ Draw eight different wired networks for the computers $C_1,C_2,C_3.$

For each state whether or not it is a connected wired network or not.

Remember, the ethernet connections are a set, so keep that in mind when you

compare to make sure they are different networks.

$($b$)$ What are the similarities and differences of wired networks and connected

wired networks?

$($c$)$ Prove or Disprove: In any wired network, the number of computers is less than

or equal to the number of ethernet connections plus $1.$