Grant: Discovery Grants Program - Individual

The title of this proposal reflects my mathematical interests generally: algebraic graph theory is my main focus, but a broader range of topics in related areas of discrete mathematics are involved. Graph theory is the mathematical study of collections of objects modelled by pairwise relationships: these could be computers connected in a network, airports connected by direct flights, people who are friends on social media, or atoms bonded in a molecule. The `algebra' involved in my work is permutation group theory, i.e. the mathematical study of symmetry, to consider graphs (i.e. networks, molecules, etc.) which are highly symmetric. As well as being of mathematical and aesthetic interest, highly symmetric graphs exist in nature (such as in the form of Carbon-60 molecules). Much of my research is focused on distance-regular graphs (DRGs), and the special case of strongly regular graphs (SRGs), which are typically highly symmetrical and/or highly structured. Both main components of this proposal involve these objects. The first theme of the proposal involves cataloguing and classifying DRGs and SRGs. Since 2016, I have been developing an online database of these graphs, which has become a valuable resource to the research community. In the case of DRGs and SRGs exhibiting a large amount of symmetry, this work has led to a number of challenging questions, both computational and theoretical, which my research program intends to address. One computational objective is to expand the tables of feasible parameters for DRGs, which have not been updated since 1989. Theoretical objectives include conjectures about DRGs/SRGs arising from certain actions of linear groups, and about Cayley graphs of elementary abelian groups, and obtaining theoretical explanations of some fascinating exceptional graphs obtained by computer search. The second theme relates distance-regular graphs to the field of graph searching. Typically, this refers to games with two sets of players who take turns moving around a graph, where one set (the "Cops") is in pursuit of the other (the "Robbers"), who in turn are trying to evade them. Recent variants of this are related to the notion of the metric dimension of a graph (efficiently locating objects using distances, similar to how a smartphone determines its geographical location), which is a subject in which I have considerable experience. While certain examples of DRGs/SRGs have been of prior interest in graph searching, there has been no systematic study; this proposal attempts to address this. Each aspect of the proposal features projects well-suited to the training of highly qualified personnel, from undergraduate students up to postdoctoral research fellows. Central to the training of HQP are considerations around equity, diversity and inclusion, which are embedded into the training program to ensure that the mathematical community benefits from a diverse range of perspectives.