Symmetries of Graphs and Networks

Symmetries in graphs and networks are closely related to the fields of group theory (more specifically, permutation group theory) and graph theory. We measure symmetry using group actions, and focus our attention on vertex-transitive graphs: graphs whose automorphism group acts transitively on the set of vertices. Although networks with high levels of symmetry do not appear often among randomly-chosen networks, they are often used in practical applications. Networks modelled on vertex-transitive graphs have been shown to be very “good” in their balance of cost (measured by the degree of each vertex in the network) against performance (how easy they are to disconnect, and the efficiency of algorithms run on them). Their symmetry often makes them relatively easy to study and understand, and has the huge advantage that “local” algorithms work globally, since the vertex-transitivity implies that all vertices hold equivalent roles within the global network. Vertex-transitive graphs also provide a beautiful context in which to study many of the general problems of graph theory; beautiful not only because of the symmetric pictures, but because of the interactions with group theory, and permutation group theory in particular. Many problems that are difficult to solve in the general context of graphs may prove tractable in this context, given our understanding of symmetry. As the field of graph theory has emerged over the past 70 years or so (since the appearance of Konig’s seminal book in 1936), symmetric graphs have thus become a very important area of study, and are often looked at closely by computer scientists and other network designers. Despite this, the number of mathematicians who study these graphs is quite small, and major gatherings have rarely (if ever) included symmetric graphs as a central focus. The field is growing rapidly; there were a relative trickle of papers on vertextransitive graphs prior to 1980, but over 1500 have been published since that year, of which over 1200 have come out since 1990, and over 500 since 2000, yet there are still few “experts. ” The goal of our workshop was to bring together these experts, both to share recent developments and techniques among themselves, and to provide a forum where younger, up-and-coming researchers can meet and learn from these established authorities. With just 20 invited participants, we had to be very selective in our choices of experts, younger researchers, and topics on which to focus, but the workshop was a great success. We invited 9 “overview” talks by experts, on topics within their expertise. These served to introduce the other participants to the latest results on major problems, and methods that have been or might be successful. They also included open problems that were suitable for working on during the free time included in the workshop schedule. We also included 7 talks that focussed on the particulars of recent work on more specific problems. Although there are many regular graph theory conferences, it is rare for them to have a more specific focus. The last major events that focussed exclusively on graph symmetry were the NATO-sponsored Summer