ULethbridge - Number Theory and Combinatorics Seminar: Connor Riddlesden

Harary and Sabidussi were the first to study the automorphisms of repeated graphs using the wreath product. Later Hemminger would go on to provide necessary and sufficient conditions for these of repeated automorphisms through the introduction of smorphisms and partitions of graphs. This notion of the wreath product tends to vary depending on the author, and comes under many different guises, including both the lexicographic product and gruppenkranz. We seek to find a generalised/alternate form of this notion and the theorem, to describe the automorphisms of any arbitrary finite connected graph G using its symmetries and tree decomposition. One method for doing this is by studying the vertex connectivity of a graph κ(G), and the associated separating sets with size equal to κ(G). Following this we can decompose the graphs into blocks based upon these separating sets and construct the tree decomposition. Then we conjecture that the automorphisms of G will be: the direct product of the automorphism of elements (excluding those in a separating set) in each block, with those automorphisms that are compatible with those in the separating set; in addition, to the semidirect product of a subset of the tree decompositions. For the case in which the vertex connectivity is one, we will present the definitive results which are currently in the process of being proved. For higher dimensional cases of vertex connectivity, we will present some of our latest findings and updated conjectures.