Let
be an injective function. For a vertex labeling f, the induced edge labeling
is defined by,
or
; then, the edge labels are distinct and are from
. Then f is called a root square mean labeling of G. In this paper, we prove root square mean labeling of some degree splitting graphs.

Graph Path Cycle Degree Splitting Graphs Root Square Mean Graphs Union of Graphs1. Introduction

The graphs considered here are simple, finite and undirected. Let denote the vertex set and denote the edge set of G. For detailed survey of graph labeling we refer to Gallian [1] . For all other standard terminology and notations we follow Harary [2] . The concept of mean labeling on degree splitting graph was introduced in [3] . Motivated by the authors we study the root square mean labeling on degree splitting graphs. Root square mean labeling was introduced in [4] and the root square mean labeling of some standard graphs was proved in [5] - [11] . The definitions and theorems are useful for our present study.

Definition 1.1: A graph with p vertices and q edge is called a root square mean graph if it is possible to label the vertices with distinct labels from in such a way that when

each edge is labeled with or, then the edge

labels are distinct and are from. In this case f is called root square mean labeling of G.

Definition 1.2: A walk in which are distinct is called a path. A path on n vertices is denoted by.

Definition 1.3: A closed path is called a cycle. A cycle on n vertices is denoted by.

Definition 1.4: Let be a graph with, where each is a set of vertices having at least two vertices and having the same degree and. The degree splitting graph of G is denoted by and is obtained from G by adding the vertices and joining to each vertex of The graph G and its degree splitting graph are given in Figure 1.

Definition 1.5: The union of two graphs and is a graph with vertex set and the edge set.

Theorem 1.6: Any path is a root square mean graph.

Theorem 1.7: Any cycle is a root square mean graph.

2. Main Results

Theorem 2.1: is a root square mean graph.

Proof: The graph is shown in Figure 2.

Let. Let the vertex set of G be where. Define a function by

The graph G and its degree splitting graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x44.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x46.png" xlink:type="simple"/></inline-formula>

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.2: Root square mean labeling of is shown in Figure 3.

Theorem 2.3: is a root square mean graph.

Proof: The graph is shown in Figure 4.

Let. Let the vertex set of G be where. Define a function by

Root square mean labeling of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x64.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x66.png" xlink:type="simple"/></inline-formula>

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.4: Root square mean labeling of is shown in Figure 5.

Theorem 2.5: is a root square mean graph.

Proof: The graph is shown in Figure 6.

Let. Let the vertex set of G be where

. Define a function by

Root square mean labeling of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x89.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x91.png" xlink:type="simple"/></inline-formula>

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.6: The labeling pattern of is shown in Figure 7.

Theorem 2.7: is a root square mean graph.

Proof: The graph is shown in Figure 8.

Let. Let the vertex set of G be where

. Define a function by

The labeling pattern of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x116.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x118.png" xlink:type="simple"/></inline-formula>

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.8: The labeling pattern of is shown in Figure 9.

Theorem 2.9: is a root square mean graph.

Proof: The graph is shown in Figure 10.

Let. Let the vertex set of G be where

.

Define a function by

Then the edges are labeled as

The labeling pattern of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x158.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x160.png" xlink:type="simple"/></inline-formula>

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.10: The root square mean labeling of is shown in Figure 11.

Theorem 2.11: is a root square mean graph.

Proof: The graph is shown in Figure 12.

Let. Let its vertex set be

where.

Define a function by

The root square mean labeling of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x190.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x192.png" xlink:type="simple"/></inline-formula>

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.12: The labeling pattern of is shown in Figure 13.

Theorem 2.13: is a root square mean graph.

Proof: The graph is shown in Figure 14.

The labeling pattern of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x206.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x208.png" xlink:type="simple"/></inline-formula>

Let. Let its vertex set be

where.

Define a function by

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

Example 2.14: The labeling pattern of is shown in Figure 15.

Theorem 2.15: is a root square mean graph.

Proof: The graph is shown in Figure 16.

Let. Let its vertex set be

The labeling pattern of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x231.png" xlink:type="simple"/></inline-formula> The graph<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x233.png" xlink:type="simple"/></inline-formula>

where.

Define a function by

Then the edges are labeled as

Then the edge labels are distinct and are from. Hence by definition 1.1, G is a root square mean graph.

The root square mean labeling of<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/5-7402719x254.png" xlink:type="simple"/></inline-formula>

Example 2.16: The root square mean labeling of is given in Figure 17.

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