M.Phil. (Mathematics) PROGRAMME CURRICULUM & SYLLABUS 2017 UGC MODEL

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KALASALINGAM UNIVERSITY (KALASALINGAM ACADEMY OF RESEARCH AND EDUCATION) (Under Section 3 of the UGC Act 1956) Anand Nagar, Krishnankoil 626 126

1 KALASALINGAM UNIVERSITY (KALASALINGAM ACADEMY OF RESEARCH AND EDUCATION) (Under Section 3 of the UGC Act 1956) Anand Nagar, Krishnankoil Srivilliputtur(via), Virudhunagar(Dt.), Tamil Nadu, INDIA M.Phil. (Mathematics) PROGRAMME CURRICULUM & SYLLABUS 2017 UGC MODEL DEPARTMENT OF MATHEMATICS KALASALINGAM UNIVERSITY

2 M.Phil. Mathematics Curriculum and Syllabus 2017 I Semester First year Course Code Course Name Course Type L T P C MLM6001 Advanced Graph Theory Theory MLM6002 Research Methodology Theory ******* Elective - I Theory TOTAL L - Lecture, T Tutorial, P Practical, C Credit II Semester Course Code Course Name Course Type L T P C MLM6099 Project Work Practical TOTAL

3 M.Phil. Mathematics Curriculum and Syllabus 2017 Electives ELECTIVES Course Course Name Course L T P C Code Type MLM6003 Introductory Combinatorics Theory MAT5008 Statistics and Computational Techniques Theory Consolidated CGPA Credits Semester Credits I Semester 15 II Semester 12 Total Credits 27

4 VISION OF THE UNIVERSITY M.Phil. (Mathematics) PROGRAMME DEPARTMENT OF MATHEMATICS KALASALINGAM UNIVERSITY (Kalasalingam Academy of Research and Education) Anand Nagar, Krishnankoil To be a Centre of Excellence of International repute in education and research MISSION OF THE UNIVERSITY To produce technically competent, socially committed technocrats and administrators through quality education and research VISION OF THE DEPARTMENT To be a global centre of excellence in mathematics for the growth of science and technology. MISSION OF THE DEPARTMENT To provide quality education and research in Mathematics through updated curriculum, effective teaching learning process. To inculcate innovative skills, team-work, ethical practices among students so as to meet societal expectations PROGRAMME OUTCOMES (PO): POs describe what students are expected to know or be able to do by the time of graduation. The Program Outcomes of M.Phil. in Mathematics are: At the end of the programme, the students will be able to: 1) Apply the concepts of graph connectivity at appropriate points of graph theory and identify the surfaces which the given graph can be embedded. 2) Use Network Flow theory to solve many real time problem such as, Hall s Marriage problem, Konig s problem, deduction of Menger s theorem, finding maximum matching in bipartite graphs. 3) Use Polya s theory to count certain configurations in the combinatorial aspects.

5 4) Apply domination theory in installing common facility in appropriate points in Town planning. Further labelling theory is used to allot radio frequency for transmission of messages. 5) Apply knowledge of Mathematics, in all the fields of learning including higher research and its extensions. PROGRAMME EDUCATIONAL OBJECTIVES (PEO): PEO 1: Technical Proficiency: Victorious in getting employment in different areas, such as industries, laboratories, Banks, Insurance Companies, Educational/Research institutions, Administrative positions, since the impact of the subject concerned is very wide. PEO 2: Professional Growth: Keep on discovering new avenues in the chosen field and exploring areas that remain conducive for research and development. PEO 3: Management Skills: Encourage personality development skills like time management, crisis management, stress interviews and working as a team. PROGRAM SPECIFIC OUTCOMES (PSO): PSO 1: To develop research level thinking in the field of pure and applied mathematics. PSO 2: To assimilate complex mathematical ideas and arguments. PSO 3: To improve your own learning and performance. PSO 4: To develop abstract mathematical thinking.

6 L T P Credit MLM ADVANCED GRAPH THEORY Pre-requisite: NIL Course Category: Core Course Type: Theory Course Objective(s): Graph Theory is an integral part of Discrete Mathematics and has applications in diversified areas such as Electrical Engineering, Computer science, Linguistics. In this course the students will be taken to frontier areas of Graph Theory such as, Caylor Color Graphs, Hamiltonian, Genus of a graph, Extremal Graph Theory, Ramsay Theory. Course Outcome(s): After completing this course, the student will be able to: CO1: Understand the concept of Connectivity and Cayley color graphs. CO2: Crossing number is used to design the safe of digital circuits. CO3: Understand the concept of region coloring and Nowhere zero k-flow. CO4: Extremal Graph Theory is used to guarantee the existence of certain graph of specified type. CO5: Study the Generalized Ramsey theory, Rainbow Ramsey numbers and the probabilistic method. Mapping of Course Outcome(s): CO / PO CO1 S CO2 S CO3 S CO4 S CO5 M S- Strong; M-Medium; L-Low SYLLABUS: UNIT I - Cayley color graphs Cayley color graphs The reconstruction problem Menger s theorem The toughness of a graph

7 UNIT II - Planar graphs Characterizations of planar graphs Hamiltonian planar graphs Crossing number and thickness The genus of a graph UNIT III - Genus of a graph 2-Cell embeddings of graphs The maximum genus of a graph Map colorings and flows Factorizations and decompositions UNIT IV - Extremal results on graphs Labelings of graphs - Turán s theorem Extremal results on graphs - Cages UNIT V - Ramsey theory Generalized Ramsey theory Rainbow Ramsey numbers The probabilistic method Random graphs Text Book G. Chartrand and L. Lesniak, Graphs and Digraphs,Fourth Edition, CRC Press, Boca Raton, (Sections 2.3, 2.4, 3.4, 3.5, 6.2, 6.3, 6.4, 7.1, 7.2, 7.3, 8.3, 9.2, 9.3, 11.1, 11.2, 11.3, 12.2, 12.3, 13.1, 13.2) References 1. F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, J.A. Bondy and U.S.R. Murty, Graph Theory with applications, North Holland, New York Douglas B. West, Introduction to Graph Theory, Second Edition, Prentice Hall of India Pvt. Ltd., New Delhi, COURSE PLAN: S. No. UNIT I: Cayley color graphs Topic No. of periods Cum. Hours 1 Cayley color graphs The reconstruction problem Menger s theorem The toughness of a graph 2 12

8 UNIT II: Planar graphs 7 Characterizations of planar graphs Hamiltonian planar graphs Crossing number and thickness The genus of a graph 2 24 UNIT III: Genus of a graph 13 2-Cell embeddings of graphs The maximum genus of a graph Map colorings and flows Factorizations and decompositions 4 36 UNIT IV: Extremal results on graphs 19 Labelings of graphs Turán s theorem Extremal results on graphs Cages 3 48 UNIT V: Ramsey theory 24 Generalized Ramsey theory Rainbow Ramsey numbers The probabilistic method Random graphs 3 60

9 L T P Credit MLM RESEARCH METHODOLOGY Pre-requisite: NIL Course Category: Core Course Type: Theory Course Objective(s): In this course the students will be trained to do problem solving in different areas of Graph Theory such as, Domination in Graphs, Labeling in Graphs, Colorings in Graphs, Matching Theory. Further the students will be motivated to take up research as their career. They will also be trained to write research papers and Ph.D. thesis. Course Outcome(s): After completing this course, the student will be able to: CO1: Understand the concepts of domination, Independence and Irredundance. CO2: Apply the concept Total dominations in real time life. CO3: Study the concepts Locating domination and Strong domination. CO4: Understand the concepts Magic squares, Antimagic squares, Magiclabeling and Antimagic labelling. CO5: Apply the concept of Total Labelling in some special classes of Graphs. Mapping of Course Outcome(s): CO / PO CO1 S CO2 M CO3 S CO4 S CO5 S S- Strong; M-Medium; L-Low SYLLABUS: UNIT I - Dominating sets in graphs Dominating sets in graphs - Hereditary and superhereditary properties -Independent sets Irredundant sets - Changing and unchanging domination. UNIT II - Total dominating sets in graphs Independent dominating set Total dominating set- Connected dominating set Dominating cliques Paired dominating sets

10 UNIT III - Locating Domination in Graphs Multiple domination Locating domination Strong and weak domination. Unit IV - Magic labeling in Graphs Preliminaries - Magic squares Antimagic squares Magic labeling Antimagic labeling- Edge-AntimagicLabelings - Edge-antimagic vertex labeling Edge-antimagic total labeling - Edge-antimagic total labeling of Cycles and Paths. Unit V Antimagic Labeling in Graphs Super Edge-Antimagic Labeling - Super edge-antimagic vertex labeling Super edge-antimagic total labeling- Super Edge-Antimagic Total Labeling of Cycles and Cycles with Chord-Super Edge-Antimagic Total Labeling of Wheels and Related Graphs - Friendship Graphs Fans Wheels Text Book 1. T.W. Haynes, S.T.Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., (Unit I, II & II: Sections 1.2, 3.1, 3.2, 3.3, 3.4, 5.1, 5.2, 5.3, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 7.1, 7.3, 7.5) 2. Martin Baca & Mirka Miller, Super Edge-Antimagic Graphs, A Wealth of Problems and Some Solutions (Unit IV & V: Chapters 1, 2, 3, 4, 5, 6, 7) COURSE PLAN: S. No. Topic UNIT I: Dominating sets in graphs No. of periods Cum. Hours 1 Dominating sets in graphs Hereditary and superhereditary properties Independent sets Irredundant sets Changing and unchanging domination 2 12 UNIT II: Total dominating sets in graphs 7 Independent dominating set Total dominating set Connected dominating set Dominating cliques Paired dominating sets 4 24

11 UNIT III: Locating Domination in Graphs 13 Multiple domination Locating domination Strong and weak domination 4 36 Unit IV: Magic labeling in Graphs 19 Preliminaries - Magic squares Antimagic squares Magic labeling Antimagic labeling- Edge-AntimagicLabelings Edge-antimagic vertex labeling Edge-antimagic total labeling Edge-antimagic total labeling of Cycles and Paths 2 48 Unit V: Antimagic Labeling in Graphs 24 Super Edge-Antimagic Labeling Super edge-antimagic vertex labeling Super edge-antimagic total labeling Super Edge-Antimagic Total Labeling of Cycles and Cycles with Chord-Super Edge Antimagic Total Labeling of Wheels and Related Graphs Friendship Graphs Fans Wheels 2 60

12 L T P Credit MLM INTRODUCTORY COMBINATORICS Pre-requisite: NIL Course Category: Core Course Type: Theory Course Objective(s): Introductory Combinatorics deals with the existence of certain configurations in a structure and when it exists it counts the number of such configurations. In this course we deal with the advanced concepts such as, Systems of distinct representatives, Dilworth s theorem and extremal set theory, The Principle of inclusion and exclusion; inversion formulae, Recursions and generating functions, Polya s theory of counting. Course Outcome(s): After completing this course, the student will be able to: CO1: Understand the Hall s Marriage theorem and its applications. CO2: Study Dilworth s theorem, Sperner s theorem, and the Erdos-Ko-Rado theorem. CO3: Study the Principle of inclusion and exclusion and solve problems using them. CO4: Analyze the concepts of Recursions and generating functions. CO5: Use Poly s theory to count certain configurations in the combinatorial aspects. Mapping of Course Outcome(s): CO / PO CO1 S CO2 S CO3 CO4 S CO5 S M S- Strong; M-Medium; L-Low SYLLABUS: Unit I : Systems of distinct representatives Bipartite graphs, P. Hall s condition, SDRs, König s theorem, Birkhoff s theorem. Unit II: Dilworth s theorem and extremal set theory Partially ordered sets, Dilworth s theorem, Sperner s theorem, Symmetric chains, the Erdos-Ko- Rado theoremflows in Networks: The Ford-Fulkerson theorem, the integrality theorem, a generalization of Birkhoff s theorem, circulations.

13 Unit III: The Principle of inclusion and exclusion; inversion formulae Inclusion-exclusion, derangements, Euler indication, Mobius function, Mobius inversion, Burnside s lemma, Probleme des ménages Unit IV: Recursions and generating functions Elementary recurrences, Catalan numbers, Counting of trees, Joyal theory, Lagrange inversion Unit V: Polya theory of counting The cycle index of a permutation group, counting orbits, weights, necklaces, the symmetric group; Stirling numbers. Text Book: Treatment and contents as in A course in Combinatorics (second edition) by J.H. Van Lint and R.M. Wilson, CUP, Contents: Unit I: Chapter 5 Unit II: Chapters 6 & 7 Unit III: Chapter 10 Unit IV: Chapter 14 Unit V: Chapter 37 References: 1. Martin Aigner, Combinatorial theory, Springer, Titu Andreescu and ZumingFeng, A Path to Combinatorics for Undergraduates, Springer (India), Miklos Bona, A walk through Combinatorics, (Second Edition), World Scientific Publ. Co., John Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, COURSE PLAN: S. No. Topic Unit I : Systems of distinct representatives No. of periods Cum. Hours 1. Bipartite graphs P. Hall s condition SDRs, König s theorem Birkhoff s theorem 3 12

14 Unit II: Dilworth s theorem and extremal set theory 5. Partially ordered sets, Dilworth s theorem Sperner s theorem, Symmetric chains The Erdos-Ko-Rado Theorem Flows In Networks: The Ford- Fulkerson Theorem The Integrality Theorem A Generalization Of Birkhoff s Theorem, Circulations 2 24 Unit III: The Principle of inclusion and exclusion; inversion formulae 10. Inclusion-exclusion, derangements Euler indication, Mobius function, Mobius inversion Burnside s lemma, Probleme des ménages 4 36 Unit IV: Recursions and generating functions 13. Elementary recurrences Catalan numbers Counting of trees Joyal theory Lagrange inversion 2 48 Unit V: Polya theory of counting 18. The Cycle Index Of A Permutation Group Counting Orbits, Weights Necklaces The Symmetric Group; Stirling Numbers 3 60

15 L T P Credit MAT STATISTICS AND COMPUTATIONAL METHODS Pre-requisite: NIL Course Category: Core Course Type: Theory Course Objective(s): To understand various concepts of Probability Theory such as, Probablity Distributions, Estimation Theory, Testing of Hypothesis, Design of Experiments, Optimatization methods. Course Outcome(s): After completing this course, the student will be able to: CO1: Understand various types of Probablity distributions such as, Binomial, Poisson, Geometric, Normal, Uniform, Exponential, Gamma and weibull distributions. CO2: Know to concept of Estimation Theory. CO3: Analysis Testing of Hypothesis using t-test, F-test and Chi-square test. CO4: Understand various designs of experiments. CO5: Know the concepts of Optimization methods and its applications. Mapping of Course Outcome(s): CO / PO CO1 S CO2 S L CO3 CO4 S CO5 S M S- Strong; M-Medium; L-Low SYLLABUS: Unit I : Probability Distributions Probability basic concepts Binomial, poisson, geometric, normal, uniform, exponential, Gamma and weibull distributions mean, variance, Moment generating functions. Unit II: Estimation Theory Estimation of parameters- Principles of least squares maximum likelihood estimation method of moments interval estimation. Unit III: Testing of Hypothesis Sampling distribution, large sample tests Mean and Proportion, Small sample tests t-test, F- test and Chi-square test- Goodness of fit Independence of attributes.

16 Unit IV: Design of Experiments Design of Experiments: Basic Designs, Factorial Design, Taguchi Techniques, ANOVA. Unit V: Optimization Methods Classical optimization methods, unconstrained minimization Univariate, Conjugate direction, gradient and variable metric methods, constrained minimization, feasible direction and projections. Text Book: Freund John, E and Miller, Irvin, Probability and Statistics for Engineering, 5 TH Edition, Prentice Hall, References: 1. Jay, L.Devore, Probability and Statistics for Engineering and Sciences, Brooks cole Publishing Company, Monterey, California, Gupta, S.C. and Kapoor, V.K, Fundamentals of Mathematical Statistics, 11 th Edition (Reprint), Sultan Chand and Sons, New Delhi, Trivedi, K.S., Probability and Statistics with Reliability, Queing and Computer Science Applications, PHI. 4. Kapur,J.N.and Saxena, H.C, mathematical Statistics, 18th Revised Edition, S.Chand & Co. Ltd., Douglasc.Montgomery, Design and analysis of experiements, John Wiley and Sons, 7 th edition, Philip J. Ross, Taguchi techniques for quality engineering, Mcgraw Hill Book Company, 2 nd edition, COURSE PLAN: S. No. Topic Unit I : Probability Distributions No. of periods Cum. Hours 1. Probability basic concepts Binomial, poisson, geometric, normal, uniform, exponential, Gamma and weibull distributions Mena and Variance Moment generating functions 3 12 Unit II: Estimation Theory 5. Estimation of parameters Principles of least squares 3 18

17 7. Maximum likelihood estimation Method of Moments Interval Estimation 2 24 Unit III: Testing of Hypothesis 10. Sampling distribution (Large sample tests) Mean and Proportion(Small sample tests) t-test, F-test and Chi-square test - Goodness of fit Independence of attributes 3 36 Unit IV: Design of Experiments 14. Design of Experiments Basic Designs Factorial Design Taguchi Techniques ANOVA Table 2 48 Unit V: Optimization Methods 19. Classical Optimization Methods Conjugate direction Unconstrained minimization and Univariate, Gradient and Variable Metric Methods Constrained Minimization Feasible direction and Projections. 2 60

A Course in Combinatorics

A Course in Combinatorics J. H. van Lint Technical Universüy of Eindhoven and R. M. Wilson California Institute of Technology H CAMBRIDGE UNIVERSITY PRESS CONTENTS Preface xi 1. Graphs 1 Terminology of