On the clique behavior of graphs of low degree

Abstract

To any simple graph \(G\), the clique graph operator \(K\) associates the graph \(K(G)\) which is the intersection graph of the maximal complete subgraphs of \(G\). The iterated clique graphs are defined by \(K^{0}(G)=G\) and \(K^{n}(G)=K(K^{n-1}(G))\) for \(n\ge 1\). If there are \(m<n\) such that \(K^{m}(G)\) is isomorphic to \(K^{n}(G)\) we say that \(G\) is convergent, otherwise, \(G\) is divergent. The first example of a divergent graph was shown by Neumann-Lara in the 1970s, and is the graph of the octahedron. In this paper, we prove that among the connected graphs with maximum degree 4, the octahedron is the only one that is divergent.