ON $F$-FACE MAGIC MEAN LABELING OF SOME DUPLICATED GRAPHS

By a graph, we mean a finite, connected, undirected planar graph without loops or multiple edges. By a planar graph, we mean that it can be drawn in a plane such that no two edges intersect. Duplication of an edge e = uv by a vertex v in a graph G is a new graph G where V (G) = V (G)∪{v} and E(G) = E(G)∪{uv, vv}. Vertex duplication of a path Pn, denoted by P̂n is formed by duplicating all the vertices of Pn, n ≥ 2. Vertex duplication of a cycle Cn, denoted by Ĉn, is formed by duplicating all the vertices of Cn, n ≥ 3, where n ≡ 0(mod 2). The middle graph M(G) is the graph whose vertex set is V (G)∪E(G) and two vertices are adjacent in M(G) if and only if either they are adjacent vertices