Abstract. The edge C4 graph E4(G) of a graph G has all the edges of Gas its
vertices, two vertices in E4(G) are adjacent if their corresponding edges in G are
either incident or are opposite edges of some C4. In this paper, characterizations
for E4(G)
being connected, complete, bipartite, tree etc are given. We have
also proved that E4(G) has no forbidden subgraph characterization. Some
dynamical behaviour such as convergence, mortality and touching number are also studied
this paper, the median and the antimedian of cographs are
discussed. It is shown that if G, and G2 are any two cographs, then there is a
cograph that is both Eulerian and Hamiltonian having Gl as its median and G2
as its antimedian. Moreover, the connected planar and outer planar cographs
are characterized and the median and antimedian graphs of connected, planar
cographs are listed.