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EVALUATION OF CERTAIN WEBER

RAMANUJAN CLASS INVARIANTS USING

MODULAR EQUATIONS

A DISSERTATION

Submitted by

KAVITHA B R

Roll No. 2012613013

in partial ful f il lment for the award of the degree

of

MASTER OF SCIENCE (APPLIED MATHEMATICS)

FACULTY OF SCIENCE AND HUMANTIES

ANNA UNIVERSITY

CHENNAI 600 025

APRIL 2014

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ANNA UNIVERSITY

CHENNAI 600 025

BONAFIDE CERTIFICATE

Certified that this dissertation titled LIARS DOMINATION IN

GRAPHS is the bonafide work of Ms. SUMITHRA R who carried out the

work under my supervision. Certified further that to the best of my knowledge

the work reported herein does not form part of any other dissertation on the basis

of which a degree or award was conferred on an earlier occasion on this or any

other candidate.

Dr. M. CHANDRASEKAR Dr. V.VIJAYALAKSHMI

Professor and Head SUPERVISOR

Department of Mathematics Assistant Professor

Anna University Department of Mathematics

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Chennai-600 025 Anna University (MIT Campus)

Chennai-600 044

ABSTRACT

In this dissertation, we study about the liars domination in graphs.

We study the necessary and sufficient condition for a set to be a liars dominating

set. We discuss about the liars domination number of complete, complete

bipartite, regular graphs and trees. We also discuss about the false alarms in liars

dominating set.

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ACKNOWLEDGEMENT

It is my pleasure and privilege to acknowledge and express my

deepest sense of gratitude to my supervisor Dr. V. VIJAYALAKSHMI,

Assistant Professor, Department of Mathematics, Anna University, MIT campus,

Chennai-600 044 for her guidance and for the various suggestions and precious

inputs which helped me to complete this dissertation successfully.

I would like to thank Dr. M. CHANDRASEKAR, Professor and

Head, Department of Mathematics, College of Engineering, Anna University,

Chennai-25 and Dr. S. SHANKAR, Coordinator, Division of Applied Science

and Humanities, Anna University, MIT Campus, Chennai 600 044 for

providing the necessary facilities for the completion of this dissertation.

I thank all my family members, seniors and my friends for their

valuable support throughout my course work.

Chennai-600 044 (SUMITHRA R)

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April 2014

TABLE OF CONTENTS

CHAPTER NO. TITLE PAGE NO.

ABSTRACT (ENGLISH) iii

ABSTRACT (TAMIL) iv

LIST OF FIGURES vii

LIST OF SYMBOLS viii

1 INTRODUCTION 1

2 LIARS DOMINATION NUMBER OF GRAPHS 6

3 LIARS DOMINATION NUMBER OF TREES 17

4 FALSE ALARMS IN LIARS DOMINATING SETS

IN GRAPHS 27

4.1 FALSE ALARMS 27

4.2 ON SENSITIVITY AND SPECIFICITY

OF DETECTORS 30

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REFERENCES 37

LIST OF FIGURES

FIGURE NO. TITLE PAGE NO.

1.1 Example of a graph with() 5

2.1 Complete graph 6

2.2 Complete bipartite graph 7

2.3 -regular graphs 7

2.4 Construction of from 14

3.1 with () ( )( ) 19

3.2 Trees of order 7 with () and () 21

4.1 The presented reporting vector is ( ),

while ()and ( ) 28

4.2 The case when and . Both and are

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location of an undesired event. 36

LIST OF SYMBOLS

- Closed neighborhood of vertex u.

- Complete bipartite graph with partite sizes mand n.

- Complete graph on nvertices.

- Cycle on nvertices.

() - Neighborhood of vertex u.

- Path on nvertices.

() - The minimum cardinality of a dominating set for .

() - The minimum cardinality of a double dominating set for .

() - The minimum cardinality of a liars dominating set for .

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