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With every connected graph G there is associated a metric space M(G) whose points are the vertices of the graph with the distance between two vertices a and b defined as zero if a = b or as the length of any shortest arc joining a and b if a ≠ b. A metric space M is called a graph metric space if there exists a graph G such that M = M (G), i.e., if there exists a graph G whose vertex set can be put in one-to-one correspondence with the points of M in such a way that the distance between every two points of M is equal to the distance between the corresponding vertices of G.

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