On Cycle Graphs

The cycle graph \(C(G)\) of a graph \(G\) has vertices which correspond to the chordless cycles of \(G\), and two vertices of \(C(G)\) are adjacent if the corresponding chordless cycles of \(G\) have at least one edge in common. If \(G\) has no cycle, then we define \(C(G)=\emptyset\), the empty graph. For an integer \(n \geq 2\), we define recursively the \(n\)-th iterated cycle graph \(C^n(G)\) by \(C^n(G)=C(C^{n-1}(G))\). We classify graphs according to their cycle graphs as follows. A graph \(G\) is \emph{cycle-vanishing} if there exists an integer \(n\) such that \(C^n(G)=\emptyset\); and \(G\) is \emph{cycle-periodic} if there exist two integers \(n\) and \(p \geq 1\) such that \(C^{n+p}(G)\cong C^n(G) \neq \emptyset\). Otherwise, \(G\) is cycle-expanding. We characterize these three types of graphs, and give some other results on cycle graphs.