Basic properties of generalized xyz-Point-Line transformation graphs

摘要

Given a graph G with vertex set V(G) = V and edge set E(G) = E, let L(G) be the line graph and G the complement of G. Let G~0 be the graph with V(G~0) = V and with no edges, G~1 the complete graph with the vertex set V, G~+ = G and G~- = G. 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 e {0,1, +, -}, the generalized xyz-Point-Line transformation graph T~(xyz)(G) of G is the graph with vertex set V(T~(xyz)(G)) = V ∪ E and the edge set E(T~(xyz)(G)) = E(G~x) ∪ 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 partsV and E if z = 1. In this paper, we obtain order, size, connectedness and diameter of generalized xyz-Point-Line transformation graphs.