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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Tree Related Analytic Mean Cordial Graph

Dr. A. Nellai Murugan1, S.Frieda Mary Robina2 1, 2 PG and Research Department of Mathematics

V.O.Chidambaram College

Abstract – Let G= (V, E) be a graph with p vertices and q edges. A Analytic Mean Cordial Labeling of a Graph G with vertex set V is a bijection from V to {-1,1} such that edge uv is assigned the label with the condition that the number of vertices labeled with -1 and the number of vertices labeled with 1 differby atmost 1 and the number of edges labeled with 1 anad the number of edges labeled with 0 differ atmost 1.The graph that admits a Analytic Mean Cordial Labelling is called Analytic Mean Cordial Graph. In this paper, we proved that tree Subdivided star n >, Tree - , Twig - , Star - , , Bistar - related graphs are Analytic Mean Cordial Graphs.

Keywords –Analytic Mean Cordial Graph, Analytic Mean Cordial Labeling. Bistar, Subdivided Star, Twig.

2000 Mathematics Subject classification 05C78.

I.INTRODUCTION

A Graph G is a finite nonempty set of object called vertices together with a set of unordered pairs of distinct vertices of G which is called edges.Each pair e = {u,v} of vertices in E is called edges or a line of G.In this paper, we proved that tree related Subdivided star

n >, Tree - , Twig - , Star - , , Bistar - graphs are Analytic Mean Cordial Graphs. For graph theory terminology, we follow [1,2,3]. Cordial Labeling and related definitions are in [4,5….31]

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

II.PRELIMINARIES

Let G = (V,G) be a graph with p vertices and q edges. A Analytic Mean Cordial Labeling of a Graph G with vertex set is a bijection from V to {-1,1} such that edge uv is assigned the label with the condition that the number of vertices labeled with -1 and the number of vertices labeled with 1 differ by atmost 1 and the number of edges labeled with 1 and the number of edges labeled with 0 differ atmost 1.

The graph that admits a Analytic Mean Cordial Labeling is called Analytic Mean Cordial Graph. In this paper, we proved that tree related Subdivided star n >, Tree -

, Twig - , Star - , , Bistar - graphs are Analytic Mean Cordial Graphs.

Definition:2.1

Subdivided star is a graph obtained as one point union of n paths of path length 2, It is denoted by n >.

Definition:2.2

Let Tr be any tree. Denote the tree obtained from Tr by considering two copies of Tr by adding an edge between them by Tr(Z) and in general the graph obtained from Tr(n-1) and Tr by adding an edge between them is denoted by .

Definition:2.3

A graph obtainrd from a path by attaching exactly two pendant edges to each internal of a path is called a Twig. It is denoted by .

Definition:2.4

A bipartite graph is a graph whose vertex set V(G) can be partitioned into two subsets and such that every edge of G has one end in and the other end in ; , is called a bipartition of G. If further,every vertex of is joined to all the vertices of , then Gis called a complete bipartite graph. The complete bipartite graph with bipartition , such that | | =

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

m and | | = n is denoted by . A complete bipartite graph or (or) is called a star.[1]

Definition:2.5

is a graph obtained by attaching root of a star at one end of and other end of is joined with each pendant vertex of .

Definition: 2.6

Bistar [3]is a graph obtained from by identifying the centre of at the end vertices of respectively.

III. MAIN RESULT

THEOREM: 3.1

Subdivided star is Analytic Mean Cordial Graph.

Proof:

Let G be

Let

Let

Define f : V(G) → {-1,1}

The vertex labeling are ,

The induced edge labeling are,

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Here,

and

Therefore, satisfies the condition

Hence, Subdivided star is Analytic Mean Cordial Graph.

For example, The Analytic Mean Cordial Graph are shown in the figure

-1 -1

1

0 0 0 0 0

0

u

1 1 1 1 1

1 1 1 1 1 1

-1 -1 -1 -1

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

figure 3.2

THEOREM:3.3

Twig - is Analytic Mean Cordial Graph.

Proof:

Let G be i

Let

Let

Define f : V(G) → {-1,1}

The vertex labeling are ,

The induced edge labeling are,

Here,

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

When n = 2m, m > 0

and

When n = 2m + 1, m > 0

and

Therefore, satisfies the conditions

Hence, Twig - is Analytic Mean Cordial Graph.

For example, The Analytic Mean Cordial Graph are shown in the figure

-1 -1

-1 1

0

1

1

1 0

-1 0 0

1

1

1 1 0

1 1 1

-1

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

figure 3.4

THEOREM:3.5

Star - is Analytic Mean Cordial Graph.

Proof:

Let G be

Let

Let

Define f : V(G) → {-1,1}

The vertex labeling are,

The induced edge labeling are,

Here,

When n = 2m, m > 0

and

When n = 2m + 1, m > 0

and

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Therefore, satisfies the conditions

Hence, Star is Analytic Mean Cordial Graph.

For example, The Analytic Mean Cordial Graph are shown in the fig

figure 3.6

THEOREM:3.7

is Analytic Mean Cordial Graph.

Proof:

Let G be

Let

Let

Define f : V(G) → {-1,1}

1

1 0 1 0 1

u

0

-1 1 -1 1 -1 1

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

The vertex labeling are,

The induced edge labeling are,

(

Here,

When n = 4m, m > 0

and

When n = 4m - 2, m > 0

and

When n = 2m + 1, m > 0

and

Therefore, satisfies the conditions

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Hence, is Analytic Mean Cordial Graph.

For example, The Analytic Mean Cordial Graph are shown in the fig

1

0

-1

1

0

figure 3.8

THEOREM:3.9

Bistar - is Analytic Mean Cordial Graph.

Proof:

Let G be

Let

Let

1

1

1 0

1 -1

1

-1

0 1

0

1

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Define f : V(G) → {-1,1}

The vertex labeling are ,

The induced edge labeling are,

Here,

When n = 2m, m > 0

and

When n = 2m + 1, m > 0

and

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Therefore, satisfies the conditions

Hence, Bistar is Analytic Mean Cordial Graph.

For example, The Analytic Mean Cordial Graph are shown in the figure

u

v

1 -1 1 -1 1

0

1 0

1 0 1 1

1

1 0 -1 1 0

0 1 0 1

-1 1 -1 1

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

figure 3.10

THEOREM:3.11

Tree - is Analytic Mean Cordial Graph.

Proof :

Let G be

Let

Let

Define f : V(G) → {-1,1}

The vertex labeling are ,

The induced edge labeling are,

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

Here,

When n = 2m , m > 0

and

When n = 2m - 1, m > 0

and

Therefore, satisfies the conditions

Hence, Tree is Analytic Mean Cordial Graph.

For example, The Analytic Mean Cordial Graph are shown in the figure

1 1

-1 0

-1 -1 1

1 0

1

0

-1 -1 1 -1 1 1 1 1

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

figure 3.12

IV. Conclusion

Graph theory has real life application with the use of vertex labeling, edge labeling and domination. We have analytic mean cordial labeling which can be used to analyze mean amount of signal.

V. References

1. Gallian.J.A,A Dynamical survey of graphs Labeling, The Electronic Journal of combinatorics. 6(2001) DS6.

2. . Harrary.F,Graph Theory, Adadison-Wesley Publishing Company Inc, USA, 1969.

3. .NellaiMurugan.A- STUDIES IN GRAPH THEORY –SOME LABELING PROBLEMS IN GRAPHS AND RELATED TOPICS,Ph.D, Thesis September 2011...

4. A. Nellai Murugan and V .Sripratha, Mean Square Cordial Labelling , International Journal of Innovative Research & Studies, ISSN 2319-9725 ,Volume 3, Issue 10Number 2 ,October 2014, PP 262-277.

5. A.Nellai Murugan and A.L Poornima ,Meanness of Special Class Of Graphs, International Journal of Mathematical Archive. ISSN 2229-5046, Vol 5 , issue 8, 2014, PP 151-158.

6. A.Nellai Murugan and A.Mathubala, Path Related Homo- cordial graphs, International Journal of

Innovative Science, Engineering & Technology , ISSN 2348-7968,Vol.2, Issue 8 ,August. 2015,PP

888-892. IF 1.50,IBI-Factor 2.33

7. A.Nellai Murugan ,V.Selva Vidhya and M Mariasingam, Results On Hetro- cordial graphs,

International Journal of Innovative Science, Engineering & Technology , ISSN 2348-7968,Vol.2,

Issue 8 ,August. 2015,PP 954-959. IF 1.50,IBI-Factor 2.33

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

8. A.Nellai Murugan , and R.Megala, Path Related Relaxed Cordial Graphs , International Journal

of Scientific Engineering and Applied Science (IJSEAS) - ISSN: 2395-3470,Volume-1, Issue-6,

September ,2015 ,PP 241-246 IF ISRA 0.217.

9. A.Nellai Murugan and J.Shiny Priyanka, Tree Related Extended Mean Cordial Graphs, International

Journal of Research -Granthaalayah, ISSN 2350-0530,Vol.3, Issue 9 ,September. 2015, PP 143-

148. I .F . 2.035(I2OR).

10. A.Nellai Murugan and S.Heerajohn, Cycle Related Mean Square Cordial Graphs, International Journal

Of Research & Development Organization – Journal of Mathematics,Vol.2, Issue 9 ,September. 2015,

PP 1-11.

11. A.Nellai Murugan and A.Mathubala, Special Class Of Homo- Cordial Graphs, International Journal

Emerging Technologies in Engineering Research, ISSN 2524-6410,Vol.2, Issue 3 ,October 2015, PP 1-

5.

12. A.Nellai Murugan and R.Megala Special Class of Relaxed Cordial Graphs ,International Journal

Emerging Technologies in Engineering Research, ISSN 2524-6410,Vol.3, Issue 1 ,October 2015, PP 11-

14.

13. A.Nellai Murugan and P. Iyadurai Selvaraj, Cycle and Armed Cap Cordial graphs, Global Scholastic

Research Journal of Multidisciplinary , ISSN 2349-9397,Volume , Issue 11, October 2015, PP 1-14.

ISRA 0.416

14. A.Nellai Murugan and J.Shinny Priyanka , Path Related Extended Mean Cordial Graphs ,International

Journal of Resent Advances in Multi- Disciplinary Research, ISSN 2350-0743,Vol.2, Issue 10 ,October

2015, PP 0836-0840. I F 1.005.

15. A.Nellai Murugan and G.Devakiruba., Divisor cordial labeling of Book and Cn @ K1,n., OUTREACH, A

Multi-Disciplinary Refereed Journal, Volume VIII, 2015, Pp. 86-92.

16. A. Nellai Murugan and P. Iyadurai Selvaraj, Path Related Cap Cordial Graphs, OUTREACH, A Multi-

Disciplinary Refereed Journal, Volume VIII, 2015, Pp. 100-106.

17. A. Nellai Murugan, and A.Meenakshi Sundari, Product Cordial Graph of Umbrella and C4 @ Sn.,

OUTREACH , A Multi-Disciplinary Refereed Journal, Volume VIII, 2015, Pp. 113 – 119.

18. A. Nellai Murugan, and V.Sripratha, Mean Square Cordial Labeling, OUTREACH, A Multi-

Disciplinary Refereed Journal, Volume VIII, 2015, Pp. 125 – 131.

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International Journal of Applied Science Engineering and Management VOL 2, ISSUE 10 ISSN 2454 -9940

19. A.Nellai Murugan and A.L.Poornima, Meanness of Special Class of Graph, OUTREACH, A Multi-

Disciplinary Refereed Journal, Volume VIII, 2015, Pp. 140 – 145.

20. A.Nellai Murugan and G.Esther, Mean Cordial Labeling of Star, Bi-Star and Wheel, OUTREACH , A

Multi-Disciplinary Refereed Journal, Volume VIII, 2015, Pp. 155 – 160.

21. A. Nellai Murugan and P. Iyadurai Selvaraj, Cycle & Armed Cap Cordial Graphs, International Journal

of Mathematical Combinatorics, ISSN 1937 1055 , Volume II, June 2016, Pp. 144-152.

22. A. Nellai Murugan and S. Frieda Mary Robina, Path Related Analytic Mean Cordial Graphs,

International Journal of Trend in Research and Development, ISSN 2394 9333 , Volume 3, issue 4

,July-August 2016, Pp. 505-508 . I F 2.010

23. A. Nellai Murugan and V.Nishanthini, Path Relatively Prime Cordial Graphs, World Journal of

Research and Review, ISSN 2455 3956 , Volume 3 Issue 2, August 2016, Pp. 74-77. I F 2.09

24. A. Nellai Murugan and X.Frietzie Fernando, Path Related Power Cordial Graphs, International Journal

of Modern Sciences and Engineering Technology , ISSN 2349-3755 , Volume 3, issue 7 ,August 2016,

Pp. 28-37 . SJIF 3.094

25. A. Nellai Murugan and R.Mahalakshmi, Path Related Sign Graphs, International Journal of Modern

Sciences and Engineering Technology , ISSN 2349-3755 , Volume 3, issue 8 ,August 2016, Pp. 06-11

. SJIF 3.094

26. A. Nellai Murugan and S .Sonia, Path Related Extended Mean Cordial Graphs, , International

Journal of Multidisciplinary Research and Development , ISSN 2349-4182 , Volume 3, issue 8

,August 2016, Pp. 312-317 . RJIF 5.72.

27. A. Nellai Murugan and X.Frietzie Fernando, Star Related Power Cordial Graphs, International Journal

of Engineering Research & Allied Sciences , ISSN 2455 -9660 , Volume 1, issue 6 ,August 2016, Pp.

24-28 . IF 1.46.

28. A. Nellai Murugan and S. Frieda Mary Robina, Cycle Related Analytic Mean Cordial Graphs,

International Journal of Emerging Technologies in Engineering Research, ISSN 2454 6410 , Volume 4,

issue 9 ,September 2016, Pp. 15-20 . I F 4.225

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29. A. Nellai Murugan and N.Roselin, Star Related Analytic Mean Square Cordial Graphs, International

Journal of Emerging Technologies in Engineering Research, ISSN 2454 6410 , Volume 4, issue 9

,September 2016, Pp. 21-24 . I F 4.225

30. A. Nellai Murugan and S .Sonia, Special Class Extended Mean Cordial Graphs, International Journal

for Innovative Research in Science & Technology , ISSN 2349-6010 , Volume 3, issue 4 ,September

2016, Pp. 51-55 . IF 3.559

31. A. Nellai Murugan and S. Frieda Mary Robina, Prism Related Analytic Mean Cordial Graphs,

International Journal of Scientific Engineering and Applied Sciences, ISSN 2395-3470 , Volume 2,

issue 9 ,September 2016, Pp. 32-45 . SJIF 3.466.