The triameter of a connected graph 
with vertex set 
and vertex count at least
3 is defined by (Das 2021)
 |
where 
is the graph distance between vertices 
and 
.
For a tree 
with 
vertices and 
leaves, Hak et al. (2022)
prove the sharp bound
 |
where 
is the floor function. Equality is attained by
a star graph whose 
arms are subdivided to
have lengths differing by at most 1.
For a connected graph that is a block graph, every triple of vertices attaining the triameter contains a pair whose graph distance is the graph diameter. Conversely, every pair attaining the graph diameter extends to a triple attaining the triameter (Hak et al. 2022).
