  The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximal
complete subgraphs) of $G$, two of which are adjacent in $kG$ if they have
non-empty intersection in $G$. We say that $G$ is clique convergent if
$k^nG\cong k^m G$ for some $n\not= m$, and that $G$ is clique divergent
otherwise.
  We completely characterise the clique convergent graphs in the class of (not
necessarily finite) locally cyclic graphs of minimum degree $\delta\ge 6$,
showing that for such graphs clique divergence is a global phenomenon,
dependent on the existence of large substructures. More precisely, we establish
that such a graph is clique divergent if and only if its universal triangular
cover contains arbitrarily large members from the family of so-called
"triangular-shaped graphs".
