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Explore one of the most intriguing topics of mathematics: the seven bridges of Kalingrat

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- 1. KONINGSBERG PROBLEM • Königsberg was a city in Prussia situated on the Pregel River (Today, the city is named Kaliningrad, and is a major industrial and commercial center of western Russia). • A river Pregel flows around the island Keniphof and then divides into two. • Seven bridges spanned the various branches of the river, as shown. • It became a tradition to try to walk around the town in a way that only crossed each bridge once, but it proved to be a difficult problem.
- 2. Determine whether it’s possible to walk across all the bridges exactly once in returning back to the starting land area. •In 1736 Euler proved that the walk was not possible to do. No Eulerian walk of the Konigsberg bridge problem since all four vertices are of odd edges. • He proved this by inventing a kind of diagram called a network, that is made up of vertices (dots where lines meet) and arcs (lines). Define the degree of a vertex to be the number of edges incident to it. •He used four dots (vertices) for the two riverbanks and the two islands. These have been marked A, B and C, D. Euler showed that there is a walk starting at any vertex, going through each edge exactly once and terminating at the start vertex iff the degree of each vertex is even. This walk is called Eulerian. •The seven lines (arcs) are the seven bridges.
- 3. C c d A g e Kneiphof D C a B f b g c d e A a D b f
- 4. • 3 bridges (arcs) join to riverbank A, and 3 join to riverbank B. 5 bridges (arcs) join to island C, and 3 join to island D. • All the vertices have an odd number of arcs, so they are called odd vertices. (An even vertex would have to have an even number of arcs joining to it).
- 5. WHY IS IT IMPOSSIBLE! • Euler's solution involved the observation that when a vertex is "visited" in the middle of the process of tracing a graph, there must be an edge coming into the vertex, and another edge leaving it and so the order of the vertex must be an even number. • This must be true for all but at most two of the vertices--the one you start at, and the one you end at and so a connected graph is traversable if and only if it has at most two vertices of odd order. Now the NETWORK graph shows that there are more than two vertices of odd order and therefore the graph cannot be traced. • • It is concluded that the desired walking tour of Königsberg is impossible.
- 6. ANY IDEAS FOR THE CRISIS…? • Suppose the citizens of Königsberg decided to build an eighth bridge, as in the diagram shown than it would be possible walking without traversing any bridge twice.(ONLY TWO VERTEX ARE ODD ORDER.) • Adding a ninth bridge to the diagram above will make the walking tour once again impossible.
- 7. GRAPH THEORY • • • • • • • • A graph, G, consists of two sets, V and E. V is a finite, nonempty set of vertices. E is set of pairs of vertices called edges. The vertices of a graph G can be represented as V(G). Likewise, the edges of a graph, G, can be represented as E(G). Graphs can be either undirected graphs or directed graphs. For a undirected graph, a pair of vertices (u, v) or (v, u) represent the same edge. For a directed graph, a directed pair <u, v> has u as the tail and the v as the head. Therefore, <u, v> and <v, u> represent different edges.
- 8. THREE SAMPLE GRAPHS 0 0 0 1 1 2 3 3 2 4 5 1 6 2 V(G1) = {0, 1, 2, 3} E(G1) = {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)} V(G2) = {0, 1, 2, 3, 4, 5, 6} E(G2) = {(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)} V(G3) = {0, 1, 2} E(G3) = {<0, 1>, <1, 0>, <1, 2>} (a) G1 (b) G2 (c) G3
- 9. VERTICES EDGES EVEN DEGREE ODD DEGREE EULER PATH 4 4 4 0 Yes 4 5 2 2 Yes 5 8 1 4 No
- 10. OUR OBSERVATIONS • The degree of vertex = No. of edges connected with a vertex • Simple path = Such a way that each vertex is traversed once. • Euler Path = Such a way that each edge is traversed once. • From above graphs it is concluded that Euler path is possible when odd degree vertex are either 0 or 2. In case of Konnisberges Bridge – all the four vertices were odd degree and hence as per above conclusion – Euler path is not possible. By adding one more bridge - two vertices are even degree and two are odd degree- Euler path is possible • •
- 11. APPLICATION OF GRAPH THEORY • • • • • • • • • • • • • GPS (Global Positioning System ) uses graph to find shortest path. Airlines uses closed paths and look for alternatives for cancelled flights. Power grids use it for transfer of electricity without overloading lines. Graphs are used to design computer networks. Analysis of electrical circuits Finding shortest routes Project planning Identification of chemical compounds Statistical mechanics Genetics Cybernetics: a branch where human beings use various types of networks: electric or mechanical Linguistics Social Sciences, and so on …
- 12. SUMMARY OF TRAVERSABLE GRAPH Number Of Odd Vertices Traversable Remarks 0 Yes Start anywhere and end back at the start vertex. 2 Yes Start at one odd and end at other odd vertex. More than 2 No A graph is a collection of vertices and edges. The order of a vertex is the number of the edges meet at that vertex. A graph is traversable if and only if it has 0 or 2 odd vertices.
- 13. Thank you

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