  A graph is Helly if every family of pairwise intersecting balls has a
nonempty common intersection. Motivated by previous work on dually chordal
graphs and graphs of bounded distance VC-dimension we prove several new results
on the complexity of computing the diameter and the radius on Helly graphs and
related graph classes.
  * First, we present algorithms which given an $n$-vertex $m$-edge Helly graph
$G$ as input, compute w.h.p. its radius and its diameter in time $\tilde{\cal
O}(m\sqrt{n})$. Our algorithms are based on the Helly property and on several
implications of the unimodality of the eccentricity function in Helly graphs:
every vertex of locally minimum eccentricity is a central vertex.
  * Then, we focus on $C_4$-free Helly graphs, which include, amongst other
subclasses, bridged Helly graphs and so, chordal Helly graphs and hereditary
Helly graphs. For the $C_4$-free Helly graphs, we present linear-time
algorithms for computing the eccentricity of all vertices. Doing so, we
generalize previous results on strongly chordal graphs to a much larger
subclass.
  * Finally, we derive from our findings on chordal Helly graphs a more general
one-to-many reduction from diameter computation on chordal graphs to either
diameter computation on split graphs or the {\sc Disjoint Set} problem.
Therefore, split graphs are in some sense the {\em only} hard instances for
diameter computation on chordal graphs. As a byproduct of our reduction the
eccentricity of all vertices in a chordal graph can be approximated in ${\cal
O}(m\log{n})$ time with an additive one-sided error of at most one, and on any
subclass of chordal graphs with constant VC-dimension the diameter can be
computed in truly subquadratic time.
  These above results are a new step toward better understanding the role of
abstract geometric properties in the fast computation of metric graph
invariants.
