International Journal of Mathematics and Mathematical Sciences

Volume 2018, Article ID 9250424, 5 pages

https://doi.org/10.1155/2018/9250424

## On Super Mean Labeling for Total Graph of Path and Cycle

^{1}Mathematics Department, Faculty of Sciences and Technology, State Islamic University of Syarif Hidayatullah, Jakarta, Indonesia^{2}Combinatorial and Applied Mathematics Research Group (CAMRG), Department of Mathematics, Faculty of Mathematics and Natural Sciences, Tadulako University, Palu, Indonesia

Correspondence should be addressed to Nur Inayah; di.ca.tkjniu@hayani.run

Received 27 April 2017; Revised 23 July 2017; Accepted 14 February 2018; Published 6 June 2018

Academic Editor: Dalibor Froncek

Copyright © 2018 Nur Inayah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be a graph with the vertex set and the edge set , respectively. By a graph we mean a finite undirected graph with neither loops nor multiple edges. The number of vertices of is called order of and it is denoted by . Let be a graph. A super mean graph on is an injection such that, for each edge in labeled by , the set forms . A graph which admits super mean labeling is called super mean graph. The* total graph ** of * is the graph with the vertex set and two vertices are adjacent whenever they are either adjacent or incident in . We have showed that graphs and are super mean, where is a path on vertices and is a cycle on vertices.

#### 1. Introduction and Preliminary Results

Let be a graph with the vertex set and the edge set , respectively. By a graph we mean a finite undirected, graph with neither loops nor multiple edges. The number of vertices of is called order of and it is denoted by . The number of edges of is called size of and it is denoted by . A graph is a graph with vertices and edges. Terms and notations not defined here are used in the sense of Harary [1].

In 2003, Somasundaram and Ponraj [2] have introduced the notion of mean labelings of graphs. Let be a graph. A graph is called a mean graph if there is an injective function from the vertices of* G *to such that when each edge is labeled with if is even and if is odd, then the resulting edge labels are distinct. Furthermore, the concept of super mean labeling was introduced by Ponraj and Ramya [3]. Let be an injection on . For each edge and an integer , the induced Smarandachely edge is defined by . Then is called a Smarandachely super labeling if . A graph that admits a Smarandachely super mean is called Smarandachely super graph. Particularly, if , we know that

Such a labeling is called a super mean labeling of if . A graph that admits a super mean labeling is called a super mean graph. Further discussions of mean and super mean labelings for some families of graph are provided in [4–10] and Gallian [11].

The* total graph * of is the graph with the vertex set and two vertices are adjacent whenever they are either adjacent or incident in . For instance, when , total graph of path is provided in Figure 1. Since the problem on super mean labeling for total graph of path and cycle are still open, the new our contributions are stated in the following sections.