Konigsberg Bridge Problem

Konigsberg Bridge ProblemThe earliest theme on represent theory seems to be by Leonard Euler,Solutio problematic ad situs pertinentis,Commentarii Academetarii Scientiarum Imperia tend Petropolitanae 8 (1736),128-140.Euler discusses whether or not it is possible to scroll around Konigsberg(later c some(prenominal)ed the Pregolya exactly once. Euler gave the conditions which are necessary to permit such a stroll. Thomas Pennyngton Kirkman (1856) and Wiliam romish Hamilton (1856) studied trips which certain sites exactly once.History of Euler roadways and star shotsAn Eulerian way of life is a highway in a chart which visits each edge exactly once in the theory graph .so, in the same way, an Eulerian circuit is an Eulerian style which starts and ends on the same flush. They were first discussing by Leohard Eular while solving the famous Seven tie of Konigsberg problem in 1736. Mathematically the problem can be stated like thisGiven the graph on the right, is it possible to construct a path (or a wheel around for example, a path starting and ending on the same height) which visits each edge exactly onceGraphs which allow the manufacture of so called Eulerian cycles are called Eulerian graphs. Euler observed that a necessary condition for the existence of Eulerian cycles is that all vertices in the graph have an even stagecoach, and that for an Eulerian path either all, or all but two (i.e., the two endpoint) vertices have an even degree this means the Konigsberg graph is not Eulerian. Carl Heierholzer published the first complete characterization of Eulerian graphs in 1873, by proving that in fact the Eulerian graphs are exactly the graphs which are connected and where every flower has an even degree.Example employ euler in our daily life is development in the teaching for set theory that widely put on in the schools. An opposite example is to visualizing file system organization.it bequeath allows files to appear in more(prenominal) than one directory in a computers file system.The history of the KonigsbergLeonhard Euler (1707-1783) is considered to have been the father of graph theory. His paper in 1736 on the heptad couples of Konigsberg is considered to have been the foundational paper in the subject.Konigsberg is a town, founded in 1256, that was originally in Prussia. After a stormy history, the town became part of Soviet Union and was renames Kaliningrad in 1946. In any event, during Eulers time the town had seven bridges (named Kramer, Schmiede, Holz, Hohe, Honig, Kottel, and Grunespanning) spanning the Pregel River. Figure 8.1 gives a simplified picture of how the bridges were originally configured (two of the bridges were later destroyed during ball War II, and two other demolished by the Russians.History of HamiltonianHamiltonian is introduced by Sir William Rowan Hamilton at 1857. He made a game called around the dry land and the originally in form of solid called dodecahedron. It has 20corners/for each c orner, it called as town. The problem started when the travel started from one city to another city along the edge to receive at city by pull ahead once arrived at one city. This is how the Hamiltonian is appearing. There is example of using Hamiltonian in life such as no-complete, n-cube and change of location salesman problem.Two types of Hamiltonian are Hamiltonian path and Hamiltonian cycleIntroductionPathPath is the chronological sequences of alternating vertices and edges. Which begin from a vertex and ended with a vertex. from each one vertex is preceded and followed by its endpoints.Simple pathSimple path is a path such that all its vertices and edges are distinct.Below is a graph that gives differences between path and simple path.Path 1 v,b,x,h,z(simple path)Path 2 u,c, w,f,y,g,x,e,d,v(path)Example of path is the way of bus direction from one destination to another destination. In other hand, simple path is a path that no complicated for example the direction from f aculty of FTSM to faculty of FUU.CycleCircle is a circular sequence of alternating vertices and edges. Each edges is preceded and followed by its endpoints.Simple cycleSimple cycle is a cycle such that all its vertices and edges are distinct.Cycle u,c,w,e,x,g,f,w,d,v,a,uSimple cyclev,b,x,g,y,f,w,c,u,a,v,Example of using cycle in life is when we travel to another place then come back to our home with using the different ways. some other example in ukm is the bus ways for example bus zone 2 will make a circle to take student and will come back to the initial location where the busy will take a rest.For simple cycle, we always see in sport, such as the tourist court for athletes running especially in event of 400 * 100 meters. Then, we also can see in power plant program that is simple cycle power plant (pp) program. It gives much benefit such as optimized design, reduced engineering costs, short lead times, increase availability and fast startups also mellow operational flexibility .Connected GraphConnected graph is a graph that there exists a path between all pairs of vertices.If a graph is a directed graph, there exist a path between vertexes to each other that in the graph, is called as strongly connected graph.The examples of disconnected graphsExample of using connected graph is use in building. For example Menara Berkembar Petronas, there is a bridge to connect the two buildings. another(prenominal) example is the bridge of Pulau Pinang. First use to connect the island and peninsular Malaysia.Example of disconnected graph is other hand than connected graph. For example the building of one employee is not connected by bridge with another employee. Next, the Island of Sipadan is not connected by a bridge with Borneo land.An Euler path in a graph is a path which traverses each edge of the graph exactly once. An Euler cycle is an Euler path which is contains cycle. If there are no loop graphs, without isolated vertices, the continuation of an Euler path impl ies the connected of the graph, since traversing every edge of such a graph requires visiting each vertex at least once.But, when the connected graph has an Euler path, one can be constructed by applying Fleurys algorithm. A connected graph has an Euler path if it has exactly zero or two vertices of odd degree. If every vertex has even degree, the graph has an Euler cycle.The definition and properties of Euler paths, cycles and graphs are valid for multigraph as well.The seven bridge of KonigsbergIn Konigsberg, Germany, a river ran through the city such that in its centre was an island, and after passing the island, the river broke into two parts. Seven bridges were built, so that the people of the city could get from one part to another part. A crude map of the centre of Konigsberg might look like thisThe people wondered whether or not, one could walk around the city in a way that would involve crossing each bridge exactly once.Degree of vertex border of degree of vertex in graph t heory is the number of edges which connected to a vertex. Degree of vertex also known as local degree. The list of all degree of vertex is called as degree sequences. One way to find the number of vertex is count the number of degree for each vertex that endpoint. An easy way is draw a circle around the vertex and count the number of edges that cross the circle. The degree of vertex can be add together or even. if the degree of vertex is even, it is known as degree vertex and the other hand, if the degree of vertex is odd, the vertex is called an odd vertex. To find out the degree of graph is by choose the largest degree of vertex. Example graph with have odd and even vertexExample degree of vertex is application of carrousel because there are many roads that connected. Either the value is odd or even. The road can be representing as edges and the roundabout as the vertex. Another example is the number of use degree of vertex in electrical pole such as the number of wire connected to the one pole.Hamiltonian pathHamiltonian path is also called as traceable path. Hamiltonian path is a path that visits each vertex exactly one and not ingeminate for each vertex in a graph. Hamiltonian graph us use to solve a problem when find a path that only visited each vertex only one in a graph.Hamiltonian cycleHamiltonian cycle is a cycle that goes through the entire city (vertex) only once for a graph. It cannot be repeated to reach a city for a one cycle except the starting and the ended city.Results of research and real world examplesGraphs can be use to represent oil flow in pipes, traffic flow on motorways, transport of pollution by rivers, groundwater movement of contamination, biochemical pathways, and the underground network.The example of Euler pathThere are many effective applications to Euler circuits and paths. Networks can be use to solve many difficult problems, like the Konigsberg Bridge problem. The can also used by mail carrier who wants to have a route where they do not retrace any of their previous steps. Other than that, Euler circuits and paths are also useful to painters, garbage collector, airplanes pilots and world navigators. Below are the examples of how Euler circuit and paths are useful in the real world.The maps that pilots use are called route maps. The route maps show the paths of the airplanes from one destination to another. Here is an example of actual route map. The centre for all travel with this airline is in Denver, Colorado. From there, we can travel to some of the major cities in the surrounding states.The Navigation below is a trip to see all different regions of the world.The above regions of the world have all been given different colors. Each region also has been given marked with a node or vertex and some (but not all) of the regions are connected with arcs.Conclusions and recommendationsAs the conclusion towards this incident project, the study of graphs and their properties is a classical subject in close to computer science department around the world. Graph Theory can be further exploited by object-oriented software engineering, taking advantage of recent research in various fields. Other than that, Graph theory is one of the top reasons to specify linear Algebra. So, all graphs (included directed, weighted, and multi-graphs) can be represented intuitive by adjency matrices, and matrix operations often end up being meaningful in terms of graph they represent. Seeing the connection between a graph and its matrix helps to understand both of them, and being able to switch back and frontward between mental models is often useful.For example, a person in many fields of modeling, are mostly easily thought of the weight graphs, and are most easily manipulated as matrices. By learning the entire graph, the student can get many benefits by it especially the computer science student. So, our recommendation towards this division in order to make the student easy to learn and improve themselves are for example, shoot the student to make a lot of exercise. Other than that, ask them to make an assignment about this topic. So that the student can search many information based on this topic and become more well-known(prenominal) and understand about graph theory