Graph of finite diameter

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This article defines a property that can be evaluated to true/false for any undirected graph, and is invariant under graph isomorphism. Note that the term "undirected graph" as used here does not allow for loops or parallel edges, so there can be at most one edge between two distinct vertices, the edge is completely described by the vertices it joins, and there can be no edge from a vertex to itself.
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Definition

Note that in the definition below, we exclude the case of the graph with zero vertices, which we trivially assume to have finite diameter.

A connected undirected graph is termed a graph of finite diameter if it satisfies the following equivalent conditions:

  1. There is a vertex whose eccentricity is finite.
  2. The eccentricity of every vertex of the graph is finite.
  3. The radius of the graph is finite.
  4. The diameter of the graph is finite.