On super mean labeling for total graph of path and cycle (Q1652930)

Summary: 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 \(f : V \rightarrow \{1,2, 3 \ldots, p + q \}\) such that, for each edge \(e = u v\) in \(E\) labeled by \(f^\ast (e) = \lceil(f (u) + f(v)) / 2\rceil\), the set \(f (V) \cup \{f^\ast (e) : e \in E \}\) forms \(\{1,2, 3, \ldots, p + q\}\). A graph which admits super mean labeling is called super mean graph. The \textit{total graph} \(T(G)\) \textit{of} \(G\) is the graph with the vertex set \(V \cup E\) and two vertices are adjacent whenever they are either adjacent or incident in \(G\). We have showed that graphs \(T(P_n)\) and \(T (C_n)\) are super mean, where \(P_n\) is a path on \(n\) vertices and \(C_n\) is a cycle on \(n\) vertices.