Figure $3$: Network for Q$3,$ Consider the network shown in Figure $3($where link IDs are shown on side of each link$).$

$2$

$1.$ How many paths connect nodes $1$ and $4?$ List down all paths and label the paths $1$ to n$.$

$2.$ Let h be vector of path flows and x is the vector of link flows. Determine the $\backslash$Delta matrix that relates

path flows to links flows using the relation x $=\backslash$Delta h$.$

$3.$ Using the $\backslash$Delta matrix, compute the link flows if the demand from $1$ to $4$ is $8$ units and the flow on each

path is equally distributed.

$4.$ Are there other feasible path flows that can result in the same link$-$flow distribution? If so$,$ give an

example.