| CODE | MAT3425 | ||||||||
| TITLE | Topics in Topological and Algebraic Graph Theory | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | The past 150 years have seen the growth and flourishing of the field of graph theory to the point where specific areas of graph theory have themselves yielded an immense body of research and results. Two of these largest areas are topological graph theory and algebraic graph theory. 1. Topological graph theory 1.1 Planes and surfaces 1.2 Embeddings in plane and on surfaces 1.3 Kuratowski’s Theorem and its generalisation 1.4 Graph minors 1.5 Colouring graphs on surfaces 1.6 Heawood map colour theorem 1.7 Crossing numbers and crossing-critical graphs 2. Algebraic graph theory 2.1 Eigenvalues of graphs (walks, labellings, cospectral graphs) 2.2 Spectral graph theory (interlacing theorem; star sets, partitions, complements) 2.3 Chromatic polynomial, rank polynomial, Tutte polynomial 2.4 Strongly regular graphs and Moore graphs 2.5 Automorphisms of graphs (Cayley graphs) Study-unit Aims: This unit addresses the topological and algebraic streams of graph theory by attempting to give students an insight of some of the most important achievements in these two areas, while simultaneously describing some of the more recent results. The proofs discussed throughout the unit would equip the students with the necessary skills and tools so that they could be able to apply similar reasoning to related problems. At the same time, the work done is intended to expose the beauty of these two areas of graph theory and their interrelationship with other branches of mathematics and beyond. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: • Know the main definitions and results in topological and algebraic graph theory. • Appreciate the implications that these results had on the development of graph theory in general. • Build on these results by applying them to more recent problems in topological and algebraic graph theory. • Analyse the interrelationships between these two areas of graph theory and other branches of mathematics. 2. Skills: By the end of the study-unit the student will be able to: • Apply a variety of techniques to solve other related problems. • Find, select and organise information gathered from different sources. • Appreciate the intrinsic beauty and wide applicability of graph theoretical results and techniques. Main Text/s and any supplementary readings: Main Textbooks • Biggs, N. (1993) ‘Algebraic Graph Theory’ (2nd Ed.). UK: Cambridge University Press. • Gross, J.L., & Tucker, T.W. (2012) ‘Topological Graph Theory’. UK: Dover Publications. • Gross, J.L., & Yellen, J. (2006) ‘Graph theory and its applications’ (2nd Ed.). Chapman & Hall/CRC. • West, D.B. (2001) ‘Introduction to Graph Theory’ (2nd Ed.). Prentice Hall. Supplementary Readings • Bondy, J.A., & Murty, U.S.R. (1976) ‘Graph Theory with applications’. New York: Mcmillan Press Ltd. • Bondy, J.A., & Murty, U.S.R. (Eds.) (1979) ‘Graph Theory and related topics’. New York: Academic Press. |
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| ADDITIONAL NOTES | Follows from: MAT2413 Leads to: MAT5417 |
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| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | John B. Gauci |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
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