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) [arrow right] 1, 2, 3, . . . , p + q} be an injection. For a vertex labeling f, the induced Smarandachely edge m-labeling f*^sub S^ for an edge e = uv, an integer m ≥ 2 is defined by. ... Then f is called a Smarandachely super m-mean labeling if f(V(G)) ∪ f*(e) : e ∈ E(G)} = 1, 2, 3, . . . , p + q}. Particularly, in the case of m = 2, we know that. ... 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^sub n^^sup (2)^]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.

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