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The clique graph $kG$ of a graph $G$ has as its vertices the cliques (maximalcomplete subgraphs) of $G$, two of which are adjacent in $kG$ if they havenon-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 divergentotherwise.  We completely characterise the clique convergent graphs in the class of (notnecessarily 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 establishthat such a graph is clique divergent if and only if its universal triangularcover contains arbitrarily large members from the family of so-called"triangular-shaped graphs".

Characterising Clique Convergence for Locally Cyclic Graphs of Minimum Degree $\delta\ge 6$