  We prove that the clique graph operator $k$ is divergent on a locally cyclic
graph $G$ (i.e. $N_G(v)$ is a circle) with minimum degree $\delta(G)=6$ if and
only if $G$ is $6$-regular. The clique graph $kG$ of a graph $G$ has the
maximal complete subgraphs of $G$ as vertices, and the edges are given by
non-empty intersections. If all iterated clique graphs of $G$ are pairwise
non-isomorphic, the graph $G$ is $k$-divergent; otherwise, it is
$k$-convergent.
  To prove our claim, we explicitly construct the iterated clique graphs of
those infinite locally cyclic graphs with $\delta\geq6$ which induce simply
connected simplicial surfaces. These graphs are $k$-convergent if the size of
triangular-shaped subgraphs of a specific type is bounded from above. We apply
this criterion by using the universal cover of the triangular complex of an
arbitrary finite locally cyclic graph with $\delta=6$, which shows our
divergence characterisation.
