Graph Theory Research Paper

Graph theory is the area of ​​mathematics that examines the characteristics of graphs. A graph is a set of points called nodes or corners, connected by lines, called arcs or edges.

The reason for choosing the word nodes and arcs or edges and corners instead of points and lines is that the edges and corners lack the usual Euclidean properties of points and lines. You can add multiple points on the same line, but an edge can only go between a maximum of two corners. The edge may go back to the same corner, when it is then called a loop. The number of edge ends that connect to the same corner called corner’s degree. It is possible that multiple edges between the same pair of vertices, which are called multiple edges. A path in a graph is a sequence of nodes that form an arc (except possibly the last).

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The origin of the Graph Theory dates back to 1735, when an article by the Swiss mathematician Leonhard Euler, presented to the Academy of St. Petersburg and published in 1741, dealt with the problem of the Seven Bridges of Königsberg. The problem was to find a path from a point that makes return to the same point in passing once and only once by each of the seven bridges of Königsberg. A path through every edge exactly once was named Eulerian path or Eulerian tour if it ends where it began. By extension, a graph admitting an Euler circuit is said Eulerian graph, which therefore constitutes the first case of property of a graph. Euler had shown that a graph is Eulerian if each vertex has an even number of edges. It is customary to refer to as Euler’s theorem, although it has been proven only ​​130 years later by the German mathematician Carl Hierholzer.

A similar problem to go through each vertex exactly once was first solved with the particular case of a knight on a chessboard that visits every square exactly once by the Arab check by theorist Al- Adli in his book Kitab ash shatranj published in 840 and lost since. This problem was studied more carefully in the XVIII century by the French mathematicians Alexandre-Théophile Vandermonde, Pierre Remond Montmort, and Abraham de Moivre. The British mathematician Thomas Kirkman studied the more general problem of the course where one cannot pass through a vertex more than once, but such a course got the name of Hamiltonian path after the Irish mathematician William Rowan Hamilton, although it does have studied only a particular case of it. Therefore, the origin of the graph theory is given to Euler because he was the first to offer a mathematical treatment of the issue, followed by Vandermonde.

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