Basic properties of generalized xyz–Point-Line transformation graphs

Given a graph G with vertex set V(G) = V and edge set E(G) = E, let L(G) be the line graph and Ḡ the complement of G. Let G0 be the graph with V(G0) = V and with no edges, G1 the complete graph with the vertex set V, G+ = G and G– = Ḡ. Let S(G) (S*(G)) be the graph with the vertex set V ∪ E such that two vertices of S(G) (S*(G)) are adjacent if and only if one corresponds to a vertex v of G and other to an edge e of G and v is incident (resp., not incident) to e in G. Given x, y, z ∈{0, 1, +, –}, the generalized xyz–Point-Line transformation graph Txyz(G) of G is the graph with vertex set V(Txyz(G)) = V ∪ E and the edge set E(Txyz(G)) = E(Gx) ∪ E(L(G))y ∪ E(W), where W = S(G) if z = +, W = S*(G) if z = –, W is the graph with V(W) = V ∪ E and with no edges if z = 0 and W is complete bipartite graph with parts V and E if z = 1. In this paper, we obtain order, size, connectedness and diameter of generalized xyz– Point-Line transformation graphs.