Eccentricity sequence of a graph of finite diameter

Definition

For a finite graph

Suppose $G$ is a finite connected undirected graph. The eccentricity sequence of $G$ is defined as follows: for each vertex of $G$, calculate the eccentricity of that vertex. We obtain a list (possibly with repetitions) of length equal to the size of the vertex set of $G$. The eccentricity sequence is obtained by sorting this list in descending order (i.e., we get a non-increasing, though not necessarily a strictly decreasing, sequence).

Note that not every non-increasing sequence arises as the eccentricity sequence of a graph.