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.
2. On Super Mean Labeling for Total Graph of Path
The theorem proposed in this section deals with the super mean labeling for total graph of path on vertices, .
Theorem 1. The total graph of path on n vertices, , is a super mean graph for all .
Proof. Let and with , , for , and for . Immediately, we have that the cardinality of the vertex set and the edge set of are and , respectively, and so . Define an injection for as follows. for . for .
And so we haveNext, we consider the following sets: It can be verified that and so is a super mean labeling of . Hence, is a super mean graph. For a simple example, the super mean labeling for total graph of path on five vertices is provided in Figure 2.
3. On Super Mean Labeling for Total Graph of Cycle
The theorem proposed in this section deals with the super mean labeling for total graph of cycle on vertices, . For illustration, total graph of cycle on vertices is provided in Figure 3.
Theorem 2. The total graph of cycle on n vertices, , is a super mean graph if either is odd and or n is even and .
Proof. Let and , where Immediately, we have that the cardinality of the vertex set and the edge set of are and , respectively, and so .
Define an injection for odd as follows: And so we haveNext, we consider the following sets:It can be verified that and so is a super mean labeling of Hence is a super mean graph for odd .
Now define an injection for even as follows:and so we have It can be verified that and so is a super mean labeling of . Hence is a super mean graph for even . For illustration, a super mean labeling for total graph of cycle on vertices is provided in Figure 4.
4. The Duality of Super Mean Labeling
Let be a graph. Given any Smarandachely super labeling on graph , the labeling defined byis also a Smarandachely super labeling of .
For the proof, since is an injection then it is follows that is also an injection on . Hence it can be verified that the set forms and so the injection is also a Smarandachely super labeling on graph . Furthermore we call that the labeling is a dual super mean labeling of .
By using the duality property above, Theorems 1 and 2, we have the following corollary.
Corollary 3. Let and be the total graph of path and cycle with vertices, respectively.(i)For all , if for and for then is a super mean labeling for .(ii)For odd if then is a super mean labeling for .(iii)For even , if then is a super mean labeling for .
5. Summary and Remarks
Here we propose new results corresponding to super mean labeling for total graph of path and cycle. This work is an effort to relate Smarandachely super labeling and its dual for . All results reported here are in total graph of path and cycle, and . In future, it is not only possible to investigate some more results corresponding to other graph families but also Smarandachely super m-mean labeling in general as well.
Disclosure
An earlier version of this paper was presented as an abstract at Distace in Graph 2016.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to express their very great appreciation to Dr. I Wayan Sudarsana, Mrs. Selvy Musdalifah, and Mrs. Nurhasanah Daeng Mangesa for their valuable and constructive suggestion during the conducting of this research work. Their willingness to give their time so generously has been very much appreciated.