Clique dynamics of locally cyclic graphs with δ ≥ 6

Abstract

We prove that the clique graph operator k is divergent on a locally cyclic graph G (i.e. NG(v) is a circle) with minimum degree δ(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 δ≥6 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 δ=6, which shows our divergence characterisation.