Transformation graph G-+-G-+-

The transformation graph G-+-G-+- of a graph G   is the graph with vertex set V(G)∪E(G)V(G)∪E(G), in which two vertices u   and vv are joined by an edge if one of the following conditions holds: (i) u,v∈V(G)u,v∈V(G) and they are not adjacent in G  , (ii) u,v∈E(G)u,v∈E(G) and they are adjacent in G, (iii) one of u   and vv is in V(G)V(G) while the other is in E(G)E(G), and they are not incident in G. In this paper, for any graph G  , we determine the connectivity and the independence number of G-+-G-+-. Furthermore, for a graph G   of order n⩾4n⩾4, we show that G-+-G-+- is hamiltonian if and only if G   is not isomorphic to any graph in {2K1+K2,K1+K3}∪{K1,n-1,K1,n-1+e,K1,n-2+K1}{2K1+K2,K1+K3}∪{K1,n-1,K1,n-1+e,K1,n-2+K1}.