Joy Morris

Although we often introduce group theory to students using groups of symmetries, we tend to move quickly away from these intuitive representations into the realm of axioms and deductions. There are plenty of good reasons for this, not least of which...

Graphical and Digraphical Regular Representations (GRRs and DRRs) are a concrete way to visualise the regular action of a group, using graphs. More precisely, a GRR or DRR on the group G is a (di)graph whose automorphism group is isomorphic to the...

Abstract I will present a history and overview of some of the work that has been done on the lexicographic product of graphs, and related generalisations. The focus of my talk will be on the automorphism groups of such graphs, and the relationship to...

A Cayley graph Cay(G;S) has the elements of G as its vertices, with g--gs if and only if s is an element of S. There is a natural colouring of the edges of any such graph, by assigning colour s to an edge if it came from the element s of S. For a...

A Cayley graph Cay(G;S) on a group G with connection set S (closed under inverses) is the graph whose vertices are the elements of G, with g adjacent to h if and only if h is in gS. If we assign a colour c(s) to each s in S so that the inverse of s...

Determining the full automorphism group of a graph is a hard problem with a long history. I will discuss some of the major results that involve finding graphs with a given automorphism group. I will then focus on circulant graphs, and describe some...

An oriented graph is a digraph with at most one arc between any pair of vertices. We say that the action of a group on a set of points is regular if it is sharply transitive; that is, there is exactly one group element mapping any point to any other...

For any finite group G, a natural question to ask is the order of the smallest possible automorphism group for a Cayley graph on G. A particular Cayley graph whose automorphism group has this order is referred to as an MRR (Most Rigid Representation)...

A Digraphical Regular Representation (DRR) for a group G is a directed graph whose full automorphism group is the regular representation of G. In 1981, Babai showed that with five small exceptions, there is at least one DRR for every finite group. A...

A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the...