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The eccentricitye(v) of a vertexv of a connected graphG is the maximum distance fromv among the vertices ofG. A nondecreasing sequencea 1,a 2, ...,a p of nonnegative integers is said to be an eccentric sequence if there exists a connected graphG of orderp whose vertices can be labelledv 1,v 2, ...,v p so thate(v i)=a i for alli. Several properties of eccentric sequences are exhibited, and a necessary and sufficient condition for a sequence to be eccentric is presented. Sequences which are the eccentricity sequences of trees are characterized. Some properties of the eccentricity sequences of self-complementary graphs are obtained. It is shown that the radius of a nontrivial self-complementary graph is two.

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