The edge \(C_k\) graph of a graph. (English) Zbl 1420.05153

Summary: For any integer \(k\geq4\), the 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. Moreover, the dynamical behavior such as convergence, periodicity, mortality and touching number of \(E_k(G)\) are studied.


05C76 Graph operations (line graphs, products, etc.)
05C75 Structural characterization of families of graphs
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