🚀 go-pugleaf

Def. Let G be a graph and let v be in V(G). Let S_v = bigsum d(u,v)
for all u not= v ( u in V(G)). G is called self-median if 
S_v = S_u for all u and v in V(G).
Def. A graph G is called self-centered if every vertex is in the center.
(The eccenticity of a vertex v is max d(u,v) for all u in V(G). 
v is in the center if v has minimum eccentricity. It is well-know that 
a tree has at most two vertices in its center.)

CONJECTURE: Every self-median graph is self-centered.
Note that every vertex transitive graph is self-median. There are 
examples of self-centered graphs that are not self-median, so the 
above inclusion would be strict. There is no known characterization of 
self-median graphs. There are non-regular self-medain graphs - 
see the recent book by Buckley and Harary "Distances in Graphs" for 
more information.
H.S. Snevily