New families of mean graphs

Let G(V, E) be a graph with p vertices and q edges. A vertex labeling of G is an assignment f : V(G) → {1, 2, 3, ... ,p + q} be an injection. For a vertex labeling f, the induced Smarandachely edge m-labeling [f.sup.*.sub.s] for an edge e = uv, an integer m ≥ 2 is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then f is called a Smarandachely super m-mean labeling if f (V(G)) ∪ {[f.sup.*] (e) : e ∈ E(G)} = {1, 2, 3, ... ,p + q}. Particularly, in the case of m = 2, we know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Such a labeling is usually called a super mean labeling. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph, particularly, super mean graph if m = 2. In this paper, we discuss two kinds of constructing larger mean graphs. Here we prove that ([P.sub.m]; [C.sub.n])m ≥ 1, n ≥ 3, ([P.sub.m]; [Q.sub.3])m ≥ 1,([P.sub.2n]; [S.sub.m])m ≥ 3, n ≥ 1 and for any n ≥ 1 ([P.sub.n]; [S.sub.1]), ([P.sub.n]; [S.sub.2]) are mean graphs. Also we establish that [[P.sub.m]; [C.sub.n]]m ≥ 1, n ≥ 3, [[P.sub.m]; [Q.sub.3]]m ≥ 1 and [[P.sub.m]; [C.sup.(2).sub.n]]m ≥ 1, n ≥ 3 are mean graphs. Key Words: Labeling, mean labeling, mean graphs, Smarandachely edge m-labeling, Smarandachely super m-mean labeling, super mean graph. AMS(2000): 05C78