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There are two main meanings of the word graph in Mathematics:
For a graph in the first sense  points and edges  we can define a distance function between vertices x and y of a graph by taking the length of the shortest path from x to y. Such a distance function defines a metric on the vertices, and thus we get a metric space. If we put nonnegative weights on the edges and define the function d(x,y) to be the minimal sum of weights taken over all possible paths from x to y we again get a metric space with the elements being the vertices of the graph.
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