Assume you have been recently hired by the Department ofTransportation (DoT) to analyze motorized vehicle traffic flows.Your initial goal is to analyze the traffic and traffic delays in alarge metropolitan area.

You choose to use a weighted graph to represent this particularscenario. Remember that a graph is a collection of nodes (orvertices) and edges. Each edge will have a correspondingweight.

Describe how you would construct such a weighted graph. What dothe nodes represent? What do the edges represent? What values wouldthe weights represent? Is this graph a tree? Justify your design.Feel free to include an example in your description.

Describe an adjacency matrix you might create to represent thisgraph. What do the rows represent? What do the columns represent?What do the entries in the matrix represent? Feel free to create anexample or create an illustration to better explain yourself.

Suppose you wish to identify the shortest travel time betweennodes on the graph. What type of path would represent such a route?Describe this path in detail.

Suppose that the DoT is considering repairing some of the roadsin the metropolitan area. But first, the DoT wishes to assess eachand every road to determine which of the roads require repair. Youhave reported to the manager of the repair project and noted thatyou can use your graphical representation and graph-based methodsto suggest the most efficient route the inspection crewcan take to inspect all the roads. Describe in detail and name thetype of the particular path on the graph that would represent thismost efficient route. Consider all aspects of travel timeundertaken by the inspection crew. Justify your answer.

Suppose that the DoT is planning to widen roads that create abottleneck in the flow of traffic. As such, you pose thisproblem as a max flow/min cut problem. You have determinedthat roads are in need of widening if the traffic flow on the roadsis at maximum capacity; these roads are essentially the bottlenecksof the road system. These bottleneck roads can be identified usingthe Max Flow/Min Cut theorem.

Using the graph representation of the road system, explain howthe Max Flow/Min Cut theorem can be used to identify the bottleneckroads.

Create an example graph one might use to analyze this network asa Max Flow problem. Include at least 8 nodes and 16 edges. Includecapacities and flows and identify the max flow and/or min cut.