  When one studies geometric properties of graphs, local finiteness is a common
implicit assumption, and that of transitivity a frequent explicit one. By
compactness arguments, local finiteness guarantees several regularity
properties. It is generally easy to find counterexamples to such regularity
results when the assumption of local finiteness is dropped. The present work
focuses on the following problem: determining whether these regularity
properties still hold when local finiteness is replaced by an assumption of
transitivity.
  After recalling the locally finite situation, we show that there are Cayley
graphs with infinite generating systems that have infinite diameter but do not
contain any infinite geodesic ray. We also introduce a notion of generalised
diameter. The generalised diameter of a graph is either an ordinal or "truly
infinite" and captures the extension properties of geodesic paths. It is a
finite ordinal if and only if the usual diameter is finite, and in that case
the two notions agree. Besides, the generalised diameter is "truly infinite" if
and only if the considered graph contains an infinite geodesic ray. We show
that there exist Cayley graphs of abelian groups of arbitrary generalised
diameter.
  Finally, we build Cayley graphs of abelian groups that have isomorphic balls
of radius R for every R but are not globally isomorphic. This enables us to
construct a non-transitive graph such that for every R and any vertices u and
v, the ball of centre u and radius R is isomorphic to that of centre v and same
radius.

Combinatorics;Group Theory