Meta-Cayley Graphs on Dihedral Groups

>Magister Scientiae - MScThe pursuit of graphs which are vertex-transitive and non-Cayley on groups has been ongoing for some time. There has long been evidence to suggest that such graphs are a very rarety in occurrence. Much success has been had in this regard with various approaches being used. The aim of this thesis is to find such a class of graphs. We will take an algebraic approach. We will define Cayley graphs on loops, these loops necessarily not being groups. Specifically, we will define meta-Cayley graphs, which are vertex-transitive by construction. The loops in question are defined as the semi-direct product of groups, one of the groups being Z₂ consistently, the other being in the class of dihedral groups. In order to prove non-Cayleyness on groups, we will need to fully determine the automorphism groups of these graphs. Determining the automorphism groups is at the crux of the matter. Once these groups are determined, we may then apply Sabidussi's theorem. The theorem states that a graph is Cayley on groups if and only if its automorphism group contains a subgroup which acts regularly on its vertex set.Chemicals Industries Education and Training Authority (CHIETA

Strong simplicity of groups and vertex - transitive graphs

Magister Scientiae - MScIn the course of exploring various symmetries of vertex-transitive graphs, we introduce the concept of quasi-normal subgroups in groups. This is done since the symmetries of vertex-transitive graphs are intimately linked to those, fait accompli, of groups. With this, we ask if the concept of strongly simple groups has a place for consideration.South Afric

Groupoids of homogeneous factorisations of graphs

Magister Scientiae - MScThis thesis is a study on the confluence of algebraic structures and graph theory. Its aim is to consider groupoids from factorisations of complete graphs. We are especially interested in the cases where the factors are isomorphic. We analyse the loops obtained from homogeneous factorisations and ask if homogeneity is reflected in the kind of loops that are obtained. In particular, we are interested to see if we obtain either groups or quasi-associative Cayley sets from these loops. November 2008.South Afric

Some meta-cayley graphs on dihedral groups

In this paper, we define meta-Cayley graphs on dihedral groups. We fully determine the automorphism groups of the constructed graphs in question. Further, we prove that some of the graphs that we have constructed do not admit subgroups which act regularly on their vertex set; thus proving that they cannot be represented as Cayley graphs on groups

Presentations for vertex-transitive graphs

We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph

Generalisations of groups and Cayley graphs

Groups are one of the most fundamental objects in Mathematics and have been generalised in many fashions. This thesis focuses on two generalisations. Within the study of groups by geometric and combinatorial group theorists, instead of thinking about groups as an algebraic object they choose to study them through their Cayley graph. This has paved the way to many simplified proofs of properties about groups. Cayley graphs are vertex transitive graphs with a regular action by a group. However, not all vertex transitive graphs have a regular action and so cannot be Cayley graphs. This is reflected in the comparable levels of knowledge about them. The first chapter in this thesis generalises the concept of a group presentation and their associated Cayley graph. We hope this will open the door for techniques from combinatorial and geometric group theory to be applied to the study of vertex transitive groups. The second is the study of groups in higher categories. Cayley pointed out that the study of groups is really just the study of symmetries. When we categorify groups into the setting of 2-categories, we study the symmetries between the symmetries given by a classical group. That is we allow the group axioms to hold only up to natural isomorphism. From this point of view we study 2-groups in the same way people studied classical groups, namely through their actions on vectors spaces. In this setting the 2-Vector spaces. The work provides an explicit formula for their characters