Study of Mean Labelings on Product Graphs

Abstract

Let G be a(p,q) graph and let: \(V(G) \rightarrow\) {O,1,…,q} be an injection. Then G is said to have a mean labeling if for each edge uv, there exists an induced injective map \(f^*:E(G) \rightarrow\) {1,2,…,q} defined by

\(f^* (uv)= {f(u) + f(v) \over 2}\) if \(f(u)+f(v)\) is even

                \(= {f(u) + f(v) + 1 \over 2}\) if \(f(u)+f(v)\) is odd

The graph G is said to be a near mean graph if the injective map f : V(G)\(\rightarrow\){1,2,…,q-1,q+1} induces \(f^*:E(G)\rightarrow\){1,2,…,q} which is also injective, defined as above. In this paper , We looked at the direct product of paths and augmented star graphs for their mean labelings and the Cartesian product of P_n and K₄ for its near- meanness in labelings.