Graphs, Designs, Codes and Groups: Topics in Algebraic Combinatorics

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The title of this proposal reflects my interests in a broad range of topics in the branch of discrete mathematics known as algebraic combinatorics, primarily in graph theory and combinatorial design theory, but also in connection with finite group theory and coding theory.******These topics have widespread appeal. Graph theory has applications as diverse as the modelling of computer networks, or organic chemical compounds such as long-chain polymers. Design theory originated in the design of statistical experiments and the scheduling of tournaments. The study of finite permutation groups is one of the oldest subjects in abstract algebra, being a mathematical abstraction of the notion of symmetry, but one which is very relevant in reducing the difficulty of computational problem solving. Coding theory is the mathematical study of the accurate transmission and storage of data. There is also a significant overlap between each of these areas.******This proposal has three components. The first part concerns graphs and groups, studying the metric dimension of graphs (efficiently locating vertices using distances, similar to how a smartphone determines its geographical location) and related parameters for permutation groups. The second part is to build an online database of distance-regular and strongly regular graphs, which should become a valuable resource for the mathematics research community. The third part concerns combinatorial design theory, developing the theory of generalized packing and covering designs, which provide a common framework in which to understand a variety of different families of combinatorial designs simultaneously, as well as having applications to areas such as coding, communications and software testing.******All three components of the proposal feature projects well-suited to the training of highly qualified personnel, from undergraduate students up to postdoctoral research fellows.**