1. 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
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Figure 1. The graph G and its degree splitting graph
.
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Then the edges are labeled as
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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
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Then the edges are labeled as
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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
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Then the edges are labeled as
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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
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Then the edges are labeled as
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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
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Then the edges are labeled as
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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
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Then the edges are labeled as
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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.
Let
. Let its vertex set be ![]()
where
.
Define a function
by
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Then the edges are labeled as
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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 ![]()
where
.
Define a function
by
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Then the edges are labeled as
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Then the edge labels are distinct and are from
. Hence by definition 1.1, G is a root square mean graph.
Example 2.16: The root square mean labeling of
is given in Figure 17.