Mathematical theory of networks. A graph consists of vertices (also called points or nodes) and edges (lines) connecting certain pairs of vertices. An edge that connects a node to itself is called a loop. In 1735 Leonhard Euler published an analysis of an old puzzle concerning the possibility of crossing every one of seven bridges (no bridge twice) that span a forked river flowing past an island. Euler's proof that no such path exists and his generalization of the problem to all possible networks are now recognized as the origin of both graph theory and topology. Since the mid-20th century, graph theory has become a standard tool for analyzing and designing communications networks, power transmission systems, transportation networks, and computer architectures.
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Visual representation of a data set or a mathematical equation, inequality, or function to show relationships or tendencies that these formulas can only suggest symbolically and abstractly. Though histograms and pie charts are also graphs, the term usually applies to point plots on a coordinate system. For example, a graph of the relationship between real numbers and their squares matches each real number on a horizontal axis with its square on a vertical axis. The resulting set of points in this case is a parabola. A graph of an inequality is usually a shaded region on one side of a curve, whose shape depends not only on the equation or inequality but on the coordinate system chosen.
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