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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.
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