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The Graph Theory

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The Graph Theory

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  1. The Graph Theory PREPARED BY: Shivansh Srivastava (130110109055) Yash Shah (130110109053) Deep Sangani (130110109050) Parth Patil (130110109040) Kunj Parikh (14EED03)

  2. Table of contents • Motivational problems • Terminologies • Theorems • History • Practical applications • Bibliography

  3. Motivational Problems 1. Old Königsberg had seven bridges. The townspeople wondered if was possible to take a walk around the town in such a way as to cross each of the seven bridges exactly once. What do you think?

  4. Second Problem 2. How many colors do we need to color a map so that every pair of countries with a border in common have different colors?

  5. Third Problem 3. Given a map of several cities and the roads between them, is it possible for a traveling salesman to visit (pass through) each of the cities exactly once?

  6. Terminologies • adjacent: Two vertices are adjacent if they are connected by an edge. • Circuit:A circuit is a path which ends at the vertex it begins (so a loop is an circuit of length one).

  7. 3. complete graph: A complete graph with n vertices is a graph with n vertices in which each vertex is connected to each of the others. Here are the first five complete graphs: 

  8. 4. A vertex is simply drawn as a node or a dot. The vertex set of G is usually denoted by V(G), or V when there is no danger of confusion. The order of a graph is the number of its vertices, i.e. |V(G)|. • 5. An edge (a set of two elements) is drawn as a line connecting two vertices, called endpoints or (less often) endvertices. An edge with endverticesx and y is denoted by xy.

  9. 6. A loop is an edge whose endpoints are the same vertex. A link has two distinct endvertices. An edge is multiple if there is another edge with the same endvertices; otherwise it is simple. The multiplicity of an edge is the number of multiple edges sharing the same end vertices; the multiplicity of a graph, the maximum multiplicity of its edges.

  10. A walk is an alternating sequence of vertices and edges, beginning and ending with a vertex, where the vertices that precede and follow an edge in the sequence are just the two end vertices of that edge. A walk is closed if its first and last vertices are the same, and open if they are different.

  11. A trail is a walk in which all the edges are distinct. A closed trail has been called a tour or circuit, but these are not universal, and the latter is often reserved for a regular subgraph of degree two. • A Hamiltonian path is a path which uses (passes through) each vertex of the graph exactly once--with the exception that it may start and stop at the same vertex.  If the path does start and stop at the same vertex, then it is a Hamiltonian Circuit.

  12. Tree: A tree is a connected subgraph of a graph having all nodes of the graph. • Co-tree: Branche which are are not on tree are called links or chords. • Rank: If there ‘n’ nodes in a graph, the rank of the graph is (n-1).

  13. Theorems • A connected graph has an Euler path (which is not a circuit) if and only if it has exactly two vertices with odd degree.

  14. Theorem:If a connected graph has n vertices, where n>2 and each vertex has degree at least n/2, then the graph has a Hamiltonian circuit. • Theorem:If a connected graph has more than two vertices, and one vertex has degree one, then it does not have a Hamiltonian circuit.

  15. History • The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory. • the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature.

  16. One of the most famous and stimulating problems in graph theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" 

  17. Practical applications • Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. Many practical problems can be represented by graphs. • In computer science, graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc.

  18. Graph-theoretic methods, in various forms, have proven particularly useful in linguistics, since natural language often lends itself well to discrete structure. • Graph theory is also used to study molecules in chemistry and physics.

  19. Bibliography • www.wikipedia.org • Circuits and Networks by U.A. Patel

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