Q$3$ Relating path and link flows $(10$ points$)$

Consider the network shown in Figure $3($where link IDs are shown on side of each link$).$

$2$

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

Let $h$ be vector of path flows and $x$ is the vector of link flows. Determine the $$ matrix that relates path flows to links flows using the relation $x=h.$

Using the $$ matrix, compute the link flows if the demand from $1$ to $4$ is $8$ units and the flow on each path is equally distributed.

Are there other feasible path flows that

can result in the same link$-$flow distribution? If so$,$ give an example.

Figure $3$: Network for Q$3$