Graphs are used to model situations where commodities are transported from one location toanother. Graph theory involves a collection of vertices/nodes and links/edges. A key example could be in modelling a case of water supply, where the pipelines could be represented as edges and vertices could be used represent water users or pipe joins. In a simple graph, each edge connects with two different vertices; and no two edges connect with the same pair of vertices. Multigraphs on the other hand can have multiple edges connecting the same two vertices.Graph theory is used to model and examine transportation networks for example;Airline networks are modeled by the use of directed multigraphs in which:Airports could be represented by vertices.Each flight could be represented by a directed edge from the vertex representing the departure airport to the vertex representing the destination airport.Road networks can be modeled using graphs where:Vertices could represent intersections.Edges could represent roads.Undirected edges could represent two-way roads.Directed edges could represent one-way roads.A graph contains two parameters which include;A set V whose elements are called vertices or nodesA set E whose elements, called edges. The graph is commonly denoted as G (V, E). a                                                        d b cUsing the graph above as an example, Vertices/nodes are: a, b, c, and d and the set of edges/lines include ab, ad, bc and bds. Some of the types of graphs in modelling transport planning include;Null graphs which are graphs with no edges.Complete graphs which are graphs where each vertex is joined to every other vertex.Cycles-these are graphs which only join the outside of the vertices.Wheels-these are graphs which add a vertex at the center.In transport planning, graph theory is most commonly used to study problems of:Routing-a case example is the one-way street problem.Networks – case examples include, the maximum flow problem, the minimum cost flow problem and the transportation problem.Routing problem and graph theoryThe one-way street problem- According to Monteiro, Robertson and Atkinson, one of the major problems cities face in trying to control the flow of traffic is trying to find out whether it would be possible to transform two-way streets into one-way streets without causing big problems. Using an example of a graph with the shape of a four-sided square, and the direction to each edge of the graph is going clockwise, then it is always possible to get from one vertex to another.                                                                        But for the case of other graphs, moving from one vertex to another is not always possible, like in the case of a triangle. The one-way street problem is solved by using Robbins theorem which states that; a graph has a strongly connected orientation if and only the graph is connected and has no bridges.   The Konigsberg bridge problem- According to Feng Xie and David Levinson, the town of Konigsberg had seven bridges and its occupants wanted to know if it was possible for one to start at some point, cross each bridge exactly once and return to the starting point. The problem can be solved by using a eulerian closed chain. In graph theory, a chain or path is eulerian in a multigraph if it uses every edge of the graph once and only once.The Chinese postman problem -The Chinese postman problem/postman problem is used to describe finding the shortest delivery route/the route with the least resistance. The process involves starting from a point or post office in a zone and returning to the same point or post office. Graph theory is used to solve this problem by building a graph where each vertex represents a street corner and each edge represents a street.The travelling salesman problem- According to Monteiro, Robertson and Atkinson, the salesman problem is used to describe a situation where a salesman would like to visit different cities, each exactly once, and return to the same point of origin, in such a way as to minimize cost. Graphs could be used to form the problem where cities are represented by vertices and edges are represented by the cost of travel. According to Monteiro, Robertson and Atkinson this problem is solved by using a Hamiltonian cycle which has minimum sum of weights. A Hamiltonian chain or path is one that uses each vertex once and only once.The shortest route problem – The shortest route problem involves finding the shortest route between two vertices or cities in a particular network. According to Monteiro, Robertson and Atkinson this problem is solved by using a directed graph network or a directed network where weights are placed to indicate arcs. If it is a small network with a few arcs, one can manually calculate the shortest route. In the case of a large network though, finding the shortest route may be very tiresome and to avoid doing it manually, an algorithm known as Dijkstra=s Algorithm is used on a directed network.

Post Author: admin


I'm Irvin!

Would you like to get a custom essay? How about receiving a customized one?

Check it out