Degree Splitting of Heronian Mean Graphs

In this paper, we prove Heronian Mean labeling of some degree splitting graphs. Already we have proved Heronian Mean labeling for some standard graphs. Here we prove that degree splitting of Path P3, Path P4, P3ʘK1, P2ʘK1,2, P2ʘK1,3, P2ʘK3 are Heronian Mean graphs.


Introduction
By a graph we mean a finite undirected graph without loops or parallel edges.For all detailed survey of graph labeling, we refer to J. A. Gallian (Gallian, 2013).For all other standard terminology and notations we follow Harary (Harary, 1988).The concept of Mean labeling was introduced in (Somasundaram & Ponraj, 2003).The concept of Harmonic Mean labeling was introduced in (Somasundaram, Ponraj, & Sandhya).The concept of Harmonic Mean labeling on Degree Splitting graph was introduced in (Sandhya, Jeyasekharan, & David).Motivated by the above results and by the motivation of the authors we study the Heronian Mean labeling on Degree Splitting graphs.Heronian Mean labeling was introduced in (Sandhya, Merly, & Deepa) and the Heronian Mean labeling of some standard graphs was proved in (Sandhya, Merly, & Deepa).
We shall make frequent references to the definitions and theorems that are useful for our present study.A Path   is a walk in which all the vertices are distinct.

Definition 1.1:
A graph G=(V,E) with p vertices and q edges is said to be a Heronian Mean graph if it is possible to label the vertices xV with distinct labels () from 1,2,…,q+1 in such a way that when each edge  =  is labeled with, then the edge labels are distinct.In this case  is called a Heronian Mean labeling of G.

Definition 1.2:
Let G=(V,E) be a graph with  =  1 ∪  2 ∪ … .∪  ∪ , Where each   is a set of vertices having atleast two vertices and  =  −∪   .The degree splitting graph of G is denoted by DS(G) and is obtained from G by adding vertices  1 ,  2 , … .  and joining   to each vertex of   (1≤  ≤ ).The graph G and its degree splitting graph DS(G) are given in figure:1.
Theorem 1.4: Any Path   is a Heronian mean graph.

Remark 1.5:
Any graph G is a subgraph of DS(G).If G has atleast two vertices, then G contains atleast two vertices of the same degree.
Hence  =  1 is the only graph such that G=DS(G).

Main Results
Theorem 2.1: ) is a Heronian mean graph.

Proof:
The graph ( 3 ) is shown in figure:2 Then the edges are labeled with Hence by definition 1.1, G is a Heronian mean graph.

Proof:
The graph ( 4 ) is shown in figure 4.
Then the edges are labeled with Hence by definition 1.1, G is a Heronian mean graph.