The edge $C_k$ graph of a graph

For any integer $k \geq 4$, the \emph{edge $C_k$ graph $E_k(G)$} of a graph $G=(V,E)$ has all edges of $G$ as it vertices, two vertices in $E_k(G)$ are adjacent if their corresponding edges in $G$ are either incident or belongs to a copy of $C_k$. In this paper, we obtained the characterizations for the edge $C_k$ graph of a graph $G$ to be connected, complete, bipartite etc. It is also proved that the edge $C_4$ graph has no forbidden subgraph characterization. Mereover, the dynamical behavior such as convergence, periodicity, mortality and touching number of $E_k(G)$ are studied.