Graphs of algebraic objects

A rack on [𝑛] can be thought of as a set of permutations (𝑓𝑥)𝑥 ∈ [𝑛] such that 𝑓(𝑥)𝑓y = 𝑓y-1𝑓x𝑓y for all 𝑥 and y; additionally, a rack such that (𝑥)𝑓𝑥 = 𝑥 for all 𝑥 is a quandle, and a quandle such that 𝑓x2 = ɩ for all 𝑥 is a kei. These objects can be represented by loopless, edge-coloured directed multigraphs on [𝑛]; we put an edge of colour y between 𝑥 and z if and only if (𝑥)𝑓y = z. This thesis studies racks and kei via these graphical representations. In 2013, Blackburn showed that for any e G 0 and 𝑛 sufficiently large, the number of isomorphism classes of kei on [𝑛] is at least 2(1/4 - e)𝑛2, while the number of isomorphism classes of racks on [𝑛] is at most 2(c+e)𝑛2 for a constant c. The main result of this thesis is that the lower bound is asymptotically correct; for any e G 0 the number of isomorphism classes of racks on [𝑛] is at most 2(1/4 + e)𝑛2, for 𝑛 sufficiently large. We also show that in almost all of these racks almost all components of the corresponding graph have size 2. Additionally, we show several more detailed results regarding kei, including a full classification of all kei in which each component has size equal to an odd prime. This thesis also contains several partial results on another topic. The commuting graph of a finite group is defined to be the graph on the non-trivial cosets of G/Z(G) where xy is an edge if and only if 𝑥 and y commute. In 2012, Hegarty and Zhelezov proposed a family of random groups and published some results on the corresponding family of random graphs; this thesis contains some generalisations and extensions of these results.