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A graph is a structure for representing relationship among objects. For example, one could imagine a graph where the objects are people and where two people are related if they know each other. One could also imagine a graph on websites where two websites are related if there is a link between them. There are an immense number of different instances of interesting graphs to consider.
More formally, a graph is a mathematical object consisting of a set of vertices, V, and a set of edges E between vertices in V.