Graph of constant finite eccentricity

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

Suppose $G$ is a connected undirected graph. We say that $G$ is a graph of constant finite eccentricity if it satisfies the following equivalent conditions:

1. $G$ is a graph of finite diameter and its radius equals its diameter.
2. The eccentricity of every vertex of $G$ is equal to a fixed finite number.