Graphs, Networks and Algorithms - 2023 entry

Graphs are a structure used to describe the underlying connectedness of a system and, therefore, have a vast range of applications from designing circuit boards to running a business efficiently. In this module, you will learn the theory of graphs and explore their practical application to solve a range of mathematical problems.

Real life problems typically involve enormous graphs, so a key theme in this module will be solving problems efficiently.  In particular, some seemingly small and simple problems are so computationally complex that, even in this era of supercomputers, it would take longer than the lifetime of the universe to find a solution.  Through analysing the effectiveness and computational complexity of algorithms, you will learn how to refine your proposed solution methods to optimise their performance.

You should be familiar with material from the core undergraduate Mathematics curriculum at Exeter, including the theory of sets and functions, analysis, linear algebra and mathematical proof.

Pre-requisites: (MTH2001 (Analysis) and MTH2002 (Algebra)) or (MTH2008 (Real Analysis) and MTH2011 (Linear Algebra)).
 

The aim of the module is to present the central results of modern graph theory together with the algorithms used to solve network problems. Whilst accurate computation is essential and will be the focus of the computer-marked quizzes, the lectures and exam will focus on a rigorous development of theory and its application in a range of guided and unseen problems. There will be a particular emphasis on studying the correctness of algorithms (i.e. how do you know it does what you want?) and their complexity (i.e. how quickly does it do it?) in the context of network problems. A project element will allow students to demonstrate their understanding of a subtopic through independent investigation.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 perform algorithms and routine computations in the theory of graphs and networks accurately;

2 state and apply key definitions and results in the theory of graphs and networks;

3 prove core theorems in graph theory;

4 translate problems into a network format for solution;

5 analyse the effectiveness and computational complexity of algorithms for solving network problems;

6 recognise problems that are computationally complex and require heuristic solution methods;

Discipline Specific Skills and Knowledge:

7 discuss and use the material from this module in the context of the wider mathematics curriculum;

8 develop independently and with minimal guidance material relevant to this topic;

Personal and Key Transferable / Employment Skills and Knowledge:

9 communicate effectively your knowledge of this topic;

10 work independently to develop your knowledge in this subject.

Topics will be chosen from:

- basic graph theory: definitions, subgraphs, Euler tours and Hamiltonian cycles, representation of graphs;

- shortest paths and spanning trees;

- flows and the max-flow-min-cut theorem;

- minimal-cost feasible-flow problems;

- the marriage theorem;

- graph colouring and applications;

- connectivity and search in graphs;

- matchings and weighted matchings;

- postman problems and the salesman problem;

- graphic embedding and planar decomposition;

- analytic graph theory and discrete differential geometry;

- electrical networks and rectangle tilings;

- additional topics will be considered depending on the interests of the student cohort.

ELE: http://vle.exeter.ac.uk

Reading list for this module: