Question $2$: Spanning and Steiner trees

$($a$)$ Prove or provide a counterexample to the following statement: If all arcs of a network have

distinct costs $($that is$,$ no two arc costs are equal$),$ then the network has a unique minimum

spanning tree.

$($b$)$ Consider the problem of the minimum cost spanning forest, in which the goal is to identify

$k$ trees that span all nodes of a network in the minimum cost. Recommend an approach to solve

this problem when $k$ is given in advance. Then, use networkx to code this approach and solve

the minimum cost spanning forest for $k=2,3$ in the network provided in the Les Mis$$rables

graph $($nx$.$les$\_$miserables$\_$graph$()).$ Specifically, do the following: i$)$ create a network based

on the Les Mis$$rables graph; ii$)$ find the weight of each edge; iii$)$ replace it with its reciprocal.

For example, Valjean and Cosette have a weight of $31,$ so that should be replaced by $\frac{1}{31}.$ This

way, you will ensure that your minimum cost spanning forest will identify trees that contain the

most closely related $($based on co$-$occurrence$)$ characters.

$($c$)$ Write code in networkx that reads the illinois.csv zip codes, randomly selects $k$ of them $($you

may try kin$\{5,10,15\})$ and then returns the minimum cost Steiner tree connecting them.

You may assume the distances between two zip codes are the Haversine distance based on their

longitudes and latitudes. You may also assume that we may only connect two zip codes by an

edge if the distance between them is below $40$ units.