The Mapping \(\overline{C}: G \rightarrow C(\overline{G})\) a New Graph Theoretic Map

The cycle graph \(C(H)\) of a graph \(H\) is the edge intersection graph of all induced chordless cycles of \(H\). We investigate iterates of the mapping \(\overline{C}: G \rightarrow C(\overline{G})\) where \(C\) denotes the map that associates to a graph its cycle graph. We call a graph \(G\) vanishing under \(\overline{C}\) if \(\overline{C^n}(G) = 0\) for some \(n\), otherwise \(G\) is called \(\overline{C}\)-persistent. We call a graph \(G\) expanding under \(\overline{C}\) if \(|\overline{C^n}(G)| \to \infty\) as \(n \to \infty\). We show that the lowest order of a \(\overline{C}\)-expanding graph is \(6\) and determine the behaviour under \(\overline{C}\) of some special graphs, including trees, null graphs, cycles and complete bipartite graphs.