Graph equations for line graphs, total graphs, middle graphs and quasi-total graphs

Abstract

Let G be a graph with vertex-set V(G) and edge-set X(G). Let L(G) and T(G) denote the line graph and total graph of G. The middle graph M(G) of G is an intersection graph Ω(F) on the vertex-set V(G) of any graph G. Let F = V′(G) ∪ X(G) where V′(G) indicates the family of all one-point subsets of the set V(G), then M(G) = Ω(F).The quasi-total graph P(G) of G is a graph with vertex-set V(G)∪X(G) and two vertices are adjacent if and only if they correspond to two non-adjacent vertices of G or to two adjacent edges of G or to a vertex and an edge incident to it in G.In this paper we solve graph equations L(G) ≅ P(H); L(G) ≅ P(H); P(G) ≅ T(H); P(G) ≅ T(H); M(G) ≅ P(H); M(G) ≅ P(H).