DEPARTMENT OF MATHEMATICS

Syllabus

Master of Philosophy Programme

in Mathematics

2011

DEPARTMENT OF MATHEMATICS

Master of Philosophy Programme in Mathematics

Course Overview

The Master of Philosophy Program in Mathematics is being offered based on a credit

system similar to the other programmes offered by the University. The M. Phil. program

has two semesters with each semester spreading through 15 weeks. The candidate has to

submit the dissertation within two months after the second semester ends. It can be

extended by two more months on specific request. Those who fail to submit the

dissertation within the extended time period can register for a further extension of four

more months with a payment of 50% of the course fee. There will be only two repeat

chances (within three years after registration) for the course work papers and no

revaluation of papers at any stage of the program. The time taken from the admission till

the submission of the dissertation shall be considered as the duration of the M. Phil.

Program.

This programme is aimed at developing students into mature researchers and

preparing them for higher research degree and teaching.

Department Goal:

To provide all students with training in mathematics that will serve as part of

foundation for research and teaching.

Course Content

The course content for the first semester is

Marks credits Duration

1) General Research Methodology 100 4 60 hrs

2) Specific Research Methodology 100 4 60 hrs

The course content for the second semester is

Marks credits Duration

1) Paper 1 100 4 60 hrs

2) Paper 2 100 4 60 hrs

Dissertation & Viva-Voce 200 8

The Dissertation work includes presentation on the project proposal (50 Marks),

double valuation of the final dissertation (100 marks), and viva-voce examination

(50 Marks).

Assessment of course work

Each paper of the semester will be assessed upon 100 marks (Continuous Internal

Assessment (CIA) – 45 marks, Attendance –5 marks and End Semester Examination - 50

marks).

The internal assessment (comprising various components such as seminar, literature

survey, presentation, class test and so on) should be done periodically and the CIA marks

should be sent to the HOD and a copy forwarded to the General Research Coordinator as

per the following guidelines:

CIA 1: 10 Marks assessment before the 2nd

Month

CIA 2: 10 Marks assessment before the 3rd

Month

CIA 3: 25 Marks assessment before the 4th

Month

The HOD will hand over the consolidated CIA marks, before the End Semester

Examination to the Controller of Examinations, in the format as per the requirement of

the office of Examinations.

Students who fail to complete CIA requirements on the specified date may be given

another chance to repeat the CIA, before the next CIA, at the discretion of the teacher and

with the consent of the Coordinator.

At the end of each semester there shall be an end semester examination for each

paper/elective. The design/pattern of the questions and question papers need not be the

same for all disciplines. However, the design/pattern shall be approved by the Deans.

The Maximum marks for each end semester examination will be 100 and the duration is

three hours.

Two sets of independent question papers for each subject, completely sealed, should be

sent to the COE through the Coordinator. The question paper should reach the

coordinator at least 15 days in advance.

There is no minimum mark required for CIA. The minimum mark to pass in ESE for each

paper is 50%. The minimum mark to pass in each paper is 50% aggregate of CIA marks

and ESE marks.

In case a candidate fails due to low marks in CIA, he/she can re-register for that subject

with a payment of required fee and complete the CIA requirements, by attending the

classes along with the other candidates as directed by the coordinator.

If a candidate fails due to low marks in ESE, he/she can appear for the subject by paying

the prescribed fee, along with the other candidates as scheduled by the office of

examinations.

Each candidate shall work under the supervision of a guide. Specific guiding for the

research program/Dissertation may commence from the beginning of the II semester. The

HOD/Coordinator will allot guides to the candidates by the end of the first semester or in

the beginning of second semester depending upon the area of specialization.

Submission of Dissertation

The title page of dissertation, contents etc. should strictly conform to the format as

prescribed by the university and the dissertation (all copies) should carry a declaration by

the candidate and certificate duly signed and issued by the guide. The dissertation should

be hard bound.

The candidates will be granted a maximum period of six months, after completing the

course to submit the dissertation.

The M.Phil. dissertation will not be accepted for assessment, unless the candidate has

paid the prescribed fees.

The candidate shall submit five hard bound copies and a soft copy (CD) of his/her

dissertation work for assessment.

Adjudication of the M. Phil Dissertation

The dissertation submitted by the candidate under the guidance of the guide will be

assessed by two experts (one internal and one external).

The candidates also have to appear for final viva-voce. Assessment based on the viva-

voce and the dissertation, along with the assessment of theory papers of both I & II

semesters will be considered to declare the results.

The candidates will be provided with marks card and a degree certificate. The grade

points and the class obtained will be entered along with the marks on the marks card.

Cancellation M.Phil. Admission

The admission of the candidate will be cancelled under the following circumstances.

1) Fails to secure 85% attendance.

2) Fails to submit the documents/ requirements related to internal assessment.

3) Does not pay the course fee within the stipulated time.

4) Fails to submit the dissertation within the stipulated time.

Course structure

I Semester

II Semester

(Electives: Choose any one – each elective has two papers)

Paper Code Subjects Hrs./

week

Marks Credit

Elective 1. Fluid Mechanics

RMT 231a 1. Differential Equations and

Computational Methods

4 100 4

RMT 232a 2. Advanced Fluid Mechanics 4 100 4

Elective 2. Riemannian Geometry

RMT 231b 1. Submanifolds of Riemannian

Manifolds

4 100 4

RMT 232b 2. Riemannian Geometry 4 100 4

Elective 3. Graph Theory

RMT 231c 1. Computational Graph Theory 4 100 4

RMT 232c 2. Advanced Graph Theory 4 100 4

Total 8 200 8

Paper Code Subjects Hrs. /

week

Marks Credit

RMT 131 1. General Research Methodology 4 100 4

RMT 132 2. Mathematical Analysis 4 100 4

Total 8 200 8

DISSERTATION

Components Marks Credit

Presentation on the research

proposal

50 2

Double valuation of the

dissertation

100 4

Viva-Voce examination 50 2

Total 200 8

Modular Objective:

SEMESTER I: (General papers)

In the first semester, students are offered two theory papers, viz, General Research

Methodology and Specific Research Methodology (Mathematical analysis). The modular

objectives of these papers are :

RMT 131: General Research Methodology

The research methodology module is intended to assist students in planning and

carrying out research projects. The students are exposed to the principles, procedures and

techniques of implementing a research project.

RMT 132: Mathematical Analysis

The objective of this paper is to help students understand the principal concepts such

as convergence and divergence of improper integrals, Lebesgue Measure, Lebegue

integral and multivariate Calculus.

SEMESTER II: (Electives)

In the second semester, three elective papers are included. Students are supposed to

choose any one of the three electives. Each elective has two papers. The modular

objectives of each elective paper are :

Elective 1. Fluid Mechanics

Paper 1:

RMT 231(a): Differential Equations and Computational Methods

This course will focus on advanced concepts in both ordinary and partial

differential equations. Special emphasis is given to the computational methods.

Paper 2:

RMT 232(a): Advanced Fluid Mechanics

This paper provides a opportunity to explore various instability problems in fluid

mechanics such as Rayleigh-Benard instability, Marangoni instability, Double Diffusive

convection, convection in porous media and convective instability in non-Newtonian

fluids.

Elective 2. Riemannian Geometry

Paper1:

RMT 231(b): Submanifolds of Riemannian Manifolds

This paper focuses on the study of the submanifolds of Riemannian manifolds and

hence on hypersurfaces and various other types of submanifolds.

Paper2:

RMT 232(b): Riemannian Geometry

This paper focuses on differentiable manifolds and hence on Riemannian manifolds,

a central concept which plays an important role in several different mathematical and/or

scientific specialized areas.

Elective 3. Graph Theory

Paper1:

RMT 231(c): Computational Graph Theory

This paper concerns the principal concepts of graph theory such as enumeration

problems, networks, spectral graphs and graph complexity. This paper could be a useful

tool for technical research in the area of Graph Theory.

Paper2:

RMT 232(c): Advanced Graph Theory

This paper covers advanced topics in graph theory such as optimization of vertex

and edge coloring, matching algorithm, four color conjecture, domination theory and so

forth. This paper provides the foundations for advanced research in Graph Theory.

DISSERTATION

Major emphasis is given to the dissertation work on a chosen research problem. The

modular objective includes research proposal, presentations on the research work done,

submission of dissertation and viva-voce examination. The publications of the research

work in refereed journals and presentation of research work in national/international

conferences/symposia/seminars will be encouraged.

Course Curriculum

Semester - I (General papers)

Paper 1: RMT 131 GENERAL RESEARCH METHODOLOGY

Unit I 15 Hours

Research methodology: An introduction –meaning of research-objectives of research-

motivation in research –types of research- research approaches-significance of research-

research methods versus methodology-research and scientific method-importance of

knowing how research done-research processes-criteria of good research-defining

research problem-selecting the problem-necessity of defining the problem-technique

involved in defining a problem-Research design- meaning of research design-need for

research design-features of good design-different research design-basic principles of

experimental design

Unit II 15 Hours

Sampling Design: Measurement and Scaling Techniques- Methods of Data Collection, -

processing and Analysis of Data,- Sampling Fundamentals, Testing of Hypotheses - I

(Parametric or Standard Tests of Hypotheses), Chi-square Test, Analysis of Variance and

Covariance, Testing of Hypotheses - II (Nonparametric or Distribution - Free

Test),Multivariate Analysis Techniques

Unit III 15 Hours

Interpretation and report writing, technique of report writing-precaution in interpretation-

significance- different steps of report writing- layout of research report-oral presentation-

mechanics of writing- Exposure to writing tools like Latex/PDF, Camera Ready

Preparation

Unit IV 15 Hours

Originality in research, resources for research, Research skills, Time management, Role

of supervisor and Scholar, Interaction with subject expert, The Computer: Its Role in

Research, Case study interpretation: minimum 5 case studies

Texts and References:

C.R.Kothari, Research Methodology- Methods and Techniques, II edition, Vishwa

Prakashan Publications, New Delhi,2006

R.Pannerselvam, Research methodology, PHI 2006

Santosh Gupta, Methodology And Statistical Techniques,(Hardcover - 2000), ISBN-

3:9788171005017, 978-8171005017, Deep & Deep Publications

E. B. Wilson Jr: An Introduction to scientific research, Dover publications, Inc. New York 1990.

Ram Ahuja: Research Methods, Rawat Publications, New Delhi 2002.

Gopal Lal Jain: Research Methodology, Mangal Deep Publications, Jaipur 2003.

B. C. Nakra and K. K. Chaudhry: Instrumentation, measurement and analysis, TMH publishing

Co. Ltd., New Delhi 1985.

S. L. Mayers, Data analysis for Scientists, John Wiley & Sons, 1976.

Horaine Blaxter, Christina Hughes, Malcolm Tight How to research, Viva Books Pvt Ltd, 1999

Bell, J., Doing your research project, Viva, 1999 (NIAS)

Thomas A., Finding our fast, Vistaar, 1998 (NIAS)

Costello,P.J.M., Action research, Continuum, 2005 (NIAS)

Gilham,B., Case study research methods, Continuum 2005 (NIAS)

Kleinman,S., Emotions and fieldwork, Sage Pub., 1993 (NIAS)

Gregory,I., Ethics in research, Continuum, 2005 (NIAS)

Bennet,J., Evaluation methods in research, Continuum, 2005 (NIAS)

Morgan,D.L., Focus groups as qualitative research, Sage Pub., 1988 (NIAS)

Illingham,Jo., Giving presentations, OUP, 2003 (NIAS)

Denscombe,M., The good research guide, Viva, 1999 (NIAS)

Blaxter,L., How to research, Viva, 2002 (NIAS)

Ezzy,D., Qualitative analysis, Routledge, 2002 (NIAS)

Patton,M.Q., Qualitative evaluation and research methods, Sage Pub, 1990 (NIAS)

Kirk,J., Reliability and validity in qualitative research, Sage Pub, 1986 (NIAS)

Paper 2: RMT 132 MATHEMATICAL ANALYSIS

Unit I - Improper integrals 10 Hours

Convergence of improper integrals of the first and second kind, absolute and

conditional convergence, integral test for series, Abel’s test, Dirichlet’s test, Cauchy

Principal value.

Unit II – Abstract integration and positive Borel measures 20 Hours

Concept of measurability, simple functions, elementary properties of measures,

integration of positive functions, integration of complex functions, the role played by sets

of measure zero, the Riesz representation theorem, regularity properties of Borel

measures, Lebesgue measure, continuity properties of measurable functions.

Unit III – LP spaces 10 Hours

Convex functions and inequalities, the Lp – spaces, approximation by continuous

functions.

Unit IV – Functions of several variables 20 Hours

Linear transformation, continuity, differentiability, continuously differentiable

functions, inverse function theorem, implicit function theorem.

Texts and References:

1. Richard R. Goldberg : Methods of Real Analysis, John Wiley & Sons, 1976.

2. Walter Rudin: Real and Complex Analysis, McGraw-Hill, 1986.

3. Walter Rudin : Principles of Mathematical Analysis, McGraw-Hill, 1976.

4. Tom M. Apostol: Mathematical Analysis, Narosa, 2004.

5. H.L. Royden, Real Analysis, MacMillan, 1988.

Semester – II (Electives)

Elective I : FLUID MECHANICS

Paper 1: RMT 231(a)

DIFFERENTIAL EQUATIONS AND

COMPUTATIONAL METHODS

Unit I: 15 Hours

Non Linear Ordinary Differential Equations: Autonomous systems. The phase plane

and its phenomena. Types of critical points. Stability: Critical points and stability for

linear systems. Stability of Liapounov’s direct method. Simple critical points and non-

linear systems. Periodic solutions. The poincare Dendixson theorem. Methods of

solving the non-linear differential equations.

Unit II: 15 Hours

Computational Methods: Solution of linear system of equations by Successive Over

Relaxation Method. Numerical solutions of Parabolic partial differential equations by

implicit methods of Crank-Nicolson and Alternating Direction Implicit method.

Variational method of Galerkin and Rayleigh Ritz.

Unit III: 15 Hours

Computational Methods: Homotropy method, Perturbation method, Shooting method,

Tanh method and Modern methods of solving linear and non-linear partial differential

equations. Variational formulation of Boundary value problem. Mini project on solving

coupled partial differential equation.

Unit IV: 15 Hours

Introduction to Mathematica Software:

Numerical Computation: Numerical solution of differential equations, numerical

solution of initial and boundary value problems, numerical integration, numerical

differentiation, matrix manipulations, optimization techniques.

Graphics: Two- and three-dimensional plots, parametric plots, typesetting capabilities

for labels and text in plots, direct control of final graphics size, resolution etc.

Texts and References:

1. George F. Simmons, Differential Equations with Applications and Historical Notes,

Tata McGraw Hill, 2003.

2. Paul DuChateau and David Zachmann, Partial Differential Equations, Schaum Series,

1986.

3. Carnahan, Luther and Wilkes : Applied Numerical methods, John Wiley and Sons,

1969.

4. Jain, Iyengar and Jain, Numerical methods for Scientific and Engineering

computations, Wiley Eastern, 1993.

Paper 2 : RMT 232(a)

ADVANCED FLUID MECHANICS

Unit I: 15 Hours Shear Instability: Stability of flow between parallel shear flows - Squire’s theorem for

viscous and inviscid theory – Rayleigh stability equation – Derivation of Orr-Sommerfeld

equation assuming that the basic flow is strictly parallel.

Unit II: 15 Hours Convective Instability: Basic concepts of stability theory – Linear and Non-linear

theories – Rayleigh Benard Problem – Analysis into normal modes – Principle of

Exchange of stabilities – first variation principle – Different boundary conditions on

velocity and temperature – Equations of hydrodynamics in rotating frame of reference –

Rayleigh Benard problem with rotation / magnetic field for free-free boundaries –

Analysis into normal modes – Principle of Exchange of stabilities – first varitional

principle. Convective Instability with Surface Tension: Rayleigh – Benard –

Marangoni Problem - with rotation / Magnetic field.

Unit III: 15 Hours Porous Media: Different models to study convection problems in porous media –

Normal mode analysis – Principle of exchange of stabilities – first variation principle –

Solution for free-free boundaries – Double Diffusive convection in porous media.

Unit IV: 15 Hours Non – Newtonian Fluids: Constitutive equations of Maxwell, Oldroyd, Ostwald ,

Ostwald de waele, Reiner – Rivlin and Micropolar fluid. Weissenberg effect and Tom’s

effect. Equation of continuity, Conservation of momentum for non-Newtonian fluids.

Thermal and Solute concentrations transport. Boundary conditions on velocity,

temperature and concentration. Rayleigh - Benard convection in Jeffery and Micropolar

fluids with free-free and isothermal boundaries.

Texts and References :

1. S. Chardrasekhar, Hydrodynamic and hydrodmagnetic stability, Oxford University

Press, 2007 (RePrint).

2. Drazin and Reid, Hydrodynamic instability, Cambridge University Press, 2006.

3. Turner J.S., Buoyancy Effects in Fluids, Cambridge University Press, 1973.

4. Tritton D.J., Physical fluid Dynamics, Van Nostrand Reinhold Company, England,

1979.

5. Bhatnagar P.L.: A Lecture course on Non-Newtonian fluids, Lecture Notes,

Department of Applied Mathematics, I. I. Sc., Bangalore.

Research Papers:

1. C. M. Vest and V. S. Arpaci, Over stability of viscous elastic fluid layer heated from

below. J. F. M. , Vol. 36, Part III, p-613 (1969)

2. Siddheshwar, P. G. and Pranesh, S, Magnetoconvection in a micropolar fluid. Int. J.

Engg. Sci., 36, 1173-1181, (1998).

Elective II : RIEMANNIAN GEOMETRY

Paper 1: RMT 231(b) RIEMANNIAN GEOMETRY

Unit I 15 Hours

DIFFERENTIABLE MANIFOLDS: Introduction to Manifolds – The space of tangent

vectors at a point of Rn – related theorems – Vector fields on open subsets of R

n – Inverse

function theorem – Definition of Differentiable manifold – Examples : Euclidean plane –

Tangent space at a point on a manifold - Vector field on a manirfold – Lie algebra of the

vector fields on a manifold – Frobenius theorem.

Unit II 15 Hours

TENSOR FIELDS ON MANIFOLDS: Tensor fields – Differential Forms and Lie

Differentiation – Covariant Derivative of Tensors – Connections – Torsion Tensor –

Symmetric Connections – Curvature tensor field

Unit III 15 Hours

RIEMANNIAN MANIFOLDS: Riemannian Manifolds - Riemannian Connection –

Fundamental theorem of Riemannian Manifold – Curvature tensors – Bianchi Identities –

Sectional Curvature - Manifold of Constant Curvature.

Unit IV 15 Hours

CONTACT MANIFOLDS: Contact Structures – Contact structures in R2n+1 T3 and Rn+1

X PRn – Almost Contact Structures – Contact metric structures – Contact Metric

structures in R2n+1 - Normal Almost Contact Structures - K-contact Structures - Sasakian

manifolds – Sasakian Structures on R2n+1

Texts and References:

1. Kobayashi S. and Nomizu K.: Foundation of Differential Geometry, John Wiley and

Sons, New York, 1969.

2. Hicks N. J.: Notes on Differential Geometry, Van Nostrand, Princeton, 1965.

3. W. Boothby : An introduction to Differentiable Manifolds and Riemannian Geometry,

Academic Press, 1984.

4. David E Blair: Contact Manifolds in Riemannian Geometry, Springer-Verlag, 1976.

Paper 2: RMT 232(b)

SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS

Unit I 15 Hours

SUBMANIFOLDS OF RIEMANNIAN MANIFOLDS: Submanifolds of Riemannian

manifolds – Induced connection and second fundamental form – Equations of Gauss,

Codazzi and Ricci – Mean curvature and Gaussian curvature.

Unit II 15 Hours

UMBILICAL AND MINIMAL SUBMANIFOLDS: Totally umbilical submanifolds,

Totally geodesic submanifolds, Minimal submanifolds in Euclidean space.

Unit III 15 Hours

HYPERSURFACES: Hyper surfaces of a Riemannian manifolds – Hypersurface of Rn

– Fundamental forms on hypersurface of Rn.

Unit IV 15 Hours

SUBMANIFOLDS OF CONTACT DISTRIBUTION: Integral submanifolds of the

Contact Distribution – Integral submanifolds of odd dimensional spheres S2n+1

, S5, S

3.

Texts and References:

1. Kobayashi S. and Nomizu K.: Foundation of Differential Geometry, John Wiley and

Sons, New York, 1969.

2. Hicks N. J.: Notes on Differential Geometry, Van Nostrand, Princeton, 1965.

3. Chen B.Y.: Geometry of submanifolds, M. Dekker, New York, 1973

4. David E Blair: Contact Manifolds in Riemannian Geometry, Springer-Verlag, 1976.

Elective III : GRAPH THEORY

Paper 1: RMT 231(c)

COMPUTATIONAL GRAPH THEORY

Unit I: 15 Hours

Enumeration & Partitions: Labeled graphs. Polya’s enumeration, enumeration of trees

and graphs. Power group enumeration theorem. Solved and unsolved graphical

enumeration problems. Degree sequences partitions, necessary and sufficient conditions

for a sequence to be graphical, algorithms of degree sequence. Graphical related

problems.

Unit II: 15 Hours

Networks: Sorting and searching: Binary search, insertion sort, merge sort, radix sort,

counting sort, heap sort. Graph Algorithms for depth-first search, breadth search,

backtracking, branch-and-bound. Flows and cuts in Networks, solving max Flow

problems, Max flow-Min Cut problems, f-augmenting paths, Max flow- Min cut theorem,

Algorithms to find augmenting path, Ford, Fulkerson, Edmonds and Karp algorithm to

find Max Flow, properties of 0-1 networks.

Unit III: 15 Hours

Spectral graphs: Introduction to the laplacian and eigen values. Basic facts about the

spectrum of a graph. Eigen values of weighted graphs. Eigen values and its related

problems.

Unit IV: 15 Hours

Graph complexity: Basic concepts in complexity theory, relation between P, NP and

NP-complete. Basic NP-complete problems: 3-Satisfiability, 3-Dimensional matching,

vertex cover and clique, Hamiltonian circuit and partitions.

Texts and References:

1. F. Harary : Graph Theory, Addison -Wesley, 1969

2. G. Chartrand and Ping Zhang : Introduction to Graph Theory. McGraw Hill, 2005.

3. Alan Tucker : “Applied Combinatorics”, 4th

Ed., John Wiley and Sons, 2002.

4. J.A. Bondy and V.S.R. Murthy: Graph Theory with Applications, Macmillan,

1976.

5. J. Gross and J. Yellen : Graph Theory and its application, CRC Press LLC, Boca

Raton, Florida, 2000.

6. M.R. Garey and D.S. Johnson : Computers and Intractability: A guide to theory of

NP-complete problems. W.H. Freeman, San Francisco, 1979

7. N. Deo: Graph Theory, Prentice Hall of India, New Delhi, 1990.

8. D. Cvetkovic, M. Doob, I. Gutman and A. Torgasev : Recent results in theory of

graph spectra, Annals of Discrete Mathematics, No. 36. Elsevier Science, 1991.

Paper 2: RMT 232(c) ADVANCED GRAPH THEORY

UNIT -I 15 Hours

Vertex and edge covering, vertex and edge independence number, Gallai theorems in

terms of vertex and edge. Matching- perfect matching, augmenting paths, maximum

matching, Hall’s theorem for bipartite graphs, the personnel assignment problem,

a matching algorithm for bipartite graphs, Chinese postman problem. Factorizations,

1-factorization, 2-factorization.

UNIT –II 15 Hours

Vertex and edge coloring, the minimization problem for vertex and edge coloring,

Brook’s theorem for vertex coloring, Vizing’s theorem for edge coloring, Simple

sequential coloring algorithm, Welsh and Powel algorithm, Smallest-last sequential

coloring algorithm, color partition, four color conjecture, elementary homomorphism for

coloring, homomorphism interpolation theorem, chromatic polynomials.

UNIT-III 15 Hours

Basic concepts in distance in graphs. Eccentricity of a vertex, radius, diameter. The

radius and diameter of a self-complementary graph. The self-centered graph and the

properties of self-centered graphs. The median, central paths and other generalized

centers. Extremal distance problems, radius, small diameter, long paths and long cycles.

Metrics on graphs, geodetic graphs and distance hereditary graphs. Eccentric sequences,

distance sequences, diameter sequences, the distance distribution and other sequences.

UNIT-IV 15 Hours

Basic concepts in domination theory, total /connected/ independent domination number,

total/connected/ independent domatic number, bounds on total /connected/ independent

domination in terms of vertex, edge, diameter, degree, covering, independence and

connectivity of a graph, Neighbourhood number and independent neighbourhood number.

Bondage number, Cobondage number and Nonbondage number.

Texts and References:

1. J. A. Bondy and V. S. R. Murthy: Graph Theory with Applications, Macmillan,

1976.

2. F. Buckely and F. Harary: Distance in graphs, Addison -Wesley, Reading Mass,

1990.

3. G. Chartrand and P.Zhang : Introduction to graph Theory. McGraw-Hill, 2005.

4. J. Clark and D. A. Holton : A first look at graph theory, World Scientific, 1995.

5. J. Gross and J. Yellen: Graph Theory and its application, CRC Press LLC, Boca

Raton, Florida, 2000.

6. T.W. Haynes, S.T. Hedetneime and P. J. Slater: Fundamental of domination in

graphs, Marcel Decker, New York, 1998.

7. F. Harary : Graph Theory, Addison -Wesley, Reading Mass, 1969.

8. D. B. West : Introduction to graph theory, Prentice-Hall, 1996.