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Eccentricity of a vertex $v$ in a graph $G$ is defined as max $\{d(v,w):w\in V(G), w\ne v\}$. My question is why is the word eccentricity used, what is the reason?

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This measures in a sense how far the vertex is from "the center" of the graph (lat. ex centro). If a vertex lies on the perimeter of a graph, it will have high eccentricity, i.e. will be far away from the center. As an analogy, think of measuring exactly the same quantity for points in the unit disc. For which points is it small, for which is it big?

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