On Super Mean Labeling for Total Graph of Path and Cycle

Abstract EN

Let G(V,E) be a graph with the vertex set V and the edge set E, respectively.

By a graph G=(V,E) we mean a finite undirected graph with neither loops nor multiple edges.

The number of vertices of G is called order of G and it is denoted by p.

Let G be a (p,q) graph.

A super mean graph on G is an injection f:V→{1,2,3…,p+q} such that, for each edge e=uv in E labeled by f⁎e=fu+f(v)/2, the set fV∪{f⁎e:e∈E} forms 1,2,3,…,p+q.

A graph which admits super mean labeling is called super mean graph.

The total graph T(G) of G is the graph with the vertex set V∪E and two vertices are adjacent whenever they are either adjacent or incident in G.

We have showed that graphs T(Pn) and TCn are super mean, where Pn is a path on n vertices and Cn is a cycle on n vertices.