Super C Logarithmic Meanness of Graphs Obtained from Paths

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International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075,Volume-8, Issue-7, May 2019

Abstract: Two functions namely and is defined for a graph whose order and size as is an injective and is an induced bijective function of respectively. Then is called a Super -logarithmic mean labeling if

A graph that admits a Super -logarithmic mean labeling is called a Super -logarithmic mean graph. In this manuscript, some of the graphs like path, total graph of a path, middle graph of a path, triangular ladder, the graph for the graph the graph subdivision of and the arbitrary subdivision of admits Super -logarithmic mean labeling.

Keywords: Labeling, logarithmic mean labeling, logarithmic mean graph.

I. INTRODUCTION

In this paper, only finite, simple and undirected graphs are considered. For terminology, definitions we follow [6] and for survey [5].

The concept of geometric mean labeling was introduced and studied the geometric mean labeling of some standard graphs [1, 2]. The concept of super geometric labeling was first introduced by A. Durai Baskar et al. and studied the super geometric mean labeling of some special classes of graphs [3, 4].

Motivated by the works on super geometric mean labeling, we introduced a new type of labeling called Super -logarithmic mean labeling. The logarithmic mean of any two numbers need not be an integer. To assign the edge label as an integer based on the logarithmic mean, we may use either flooring function or ceiling function. In this paper, we consider the ceiling function of our discussion.

A vertex labeling of is an assignment

be an injection. For a vertex labeling , the induced edge labeling is defined as

Then f is called a

Super -logarithmic mean labeling if

A graph that admits a Super -logarithmic mean labeling is called a Super -logarithmic mean graph.

Revised Manuscript Received on May 07, 2019.

A.Durai Baskar, Research Scholar of Mathematics, Bharathiar University, Coimbatore - 641 046, Tamilnadu, India

S.Saratha Devi, Department of Mathematics, Mepco Schlenk

Fig. 1.1 A Super -logarithmic mean labeling of

In this paper, we have established the Super -logarithmic

meanness of the graph for square graph of a path, total graph of a path, middle graph of a path, triangular ladder, the graph subdivision of the graph and the arbitrary subdivision

of

II. MAIN RESULTS

Theorem 2.1 Any allows Super -logarithmic mean labeling.

Proof. Let the vertices of be . We define

as follows:

for belongs to

Then the induced edge labeling is as follows: for belongs to

Hence is a Super -logarithmic mean labeling of the path

Thus the is a Super -logarithmic mean graph.

Fig. 1.1 A Super -logarithmic mean labeling of Theorem 2.2 admits Super -logarithmic mean labeling, for

Proof. Let the vertices of be .We define

as follows:

Then the induced edge labeling is as follows:

for belongs to and

Super C-Logarithmic Meanness of Graphs

Obtained from Paths

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Hence, is a Super -logarithmic mean labeling of

Thus the graph is a Super -logarithmic mean graph, for

Fig. 2.2 A Super -logarithmic mean labeling of Theorem 2.3 The total graph is a Super -logarithmic mean graph, for

Proof. Let and

be the vertex set

and edge set of the path

Then and

We define

as follows:

and

for

and

Thus the graph is a Super -logarithmic mean graph,

[image:2.595.57.271.647.740.2]

for

Theorem 2.4 is a Super -logarithmic mean graph, for

Proof. Let and

be the vertex set and edge set of the path Then

We define

as

follows:

and for

Then the induced edge labeling is as follows: for

for

and for

Thus the graph is a Super -logarithmic mean graph

for

Theorem 2.5 is a Super -logarithmic mean graph, for

Proof. Let the vertex set of be

and the edge set of be

Then has vertices and edges.

We define as

follows:

for

and for

for

and for

Thus the graph is a Super -logarithmic mean graph

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[image:3.595.308.545.48.247.2]

Fig. 2.5 A Super -logarithmic mean labeling of Theorem 2.6 The graph is a Super -logarithmic

mean graph, for and

Proof. Let be the vertices of the path and

be the pendant vertices at each vertex

of the path for

Case 1. We define

as follows:

for and

for

Case 2.

We define

as follows:

for

for and

for

for and

for

Case 3.

We define

as follows:

for

for and

for

for and

for

Thus the graph is a Super

-logarithmic mean graph, for and

Fig. 2.6 A Super -logarithmic mean labeling of and

Theorem 2.7 is a Super -logarithmic mean graph, for

be the pendant vertices at each vertex of the

path for Then

We define

as

follows:

for

and

for

and

Thus the graph is a Super -logarithmic

[image:3.595.54.548.59.820.2]

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[image:4.595.307.547.38.627.2]

Theorem 2.8 is a Super -logarithmic mean graph, for

be the pendant vertices at each vertex

of the path for We define

as follows:

for

and

Thus the graph is a Super -logarithmic

mean graph, for

Theorem 2.9 is a Super -logarithmic mean graph, for

Proof. Let Let be

the vertex which divides the edge , for and

be the vertex which divides the edge for

Then

We define

as follows:

for

and

for

and

Thus the graph is a Super

-logarithmic mean graph, for

Theorem 2.10 An arbitrary subdivision of is a Super -logarithmic mean graph.

Proof. Let be an arbitrary subdivision of and let

and be the vertices of in which is the

central vertex and and are the pendant vertices of

Let the edges and of be subdivided

[image:4.595.56.277.51.144.2]

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Case 1. .

We define

as follows:

for

and for

for

and

Case 2. .

We define

as follows:

for

and for

for

and

III. CONCLUSION

Hence, is a Super -logarithmic mean labeling of Thus

[image:5.595.312.563.99.227.2]

the graph is a Super -logarithmic mean graph.

Fig. 2.10 A Super -logarithmic mean labeling of arbitrary subdivision of

REFERENCES

1. A. Durai Baskar, S. Arockiaraj and B. Rajendran, -Geometric mean labeling of some chain and thorn graphs, Kragujevac Journal of Mathematics, 37(1) (2013), 163-186.

2. A. Durai Baskar, S. Arockiaraj and B. Rajendran, Geometric meanness of graphs obtained from paths, Utilitas Mathematica, 101 (2016) 45-68.

3. A. Durai Baskar and S. Arockiaraj, Super Geometric Mean Graphs,

SUT Journal of Mathematics, 52 (2016) 97-116.

4. A. Durai Baskar and S. Arockiaraj, Further Results on Super Geometric Mean Graphs, International Journal of Mathematical Combinatorics, 4 (2015) 104-128.

5. J.A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 17(2017).