Gallai and anti-Gallai Graph Operators

The Gallai graph Γ(G) of a graph G, has the edges of G as its vertices and two distinct edges are adjacent in Γ(G) if they are incident in G, but do not span a triangle. The anti-Gallai graph Δ(G) of a graph G, has the edges of G as its vertices and two distinct edges of G are adjacent in Δ(G) if they lie on a common triangle in G. In this paper we study graphs G for which Γ(G)≅Δ(G). We also prove that, there does not exist any graph G for which Γ(Δ(G))≅Δ(Γ(G))≅H, where H is diamond-free.