## Abstract

[Truncated] A graph Γ is *self-complementary* if its complement is isomorphic to the graph itself. An isomorphism that maps Γ to its complement Γ is called a *complementing isomorphism*. The majority of this dissertation is intended to present my research results on the study of self-complementary vertex-transitive graphs. I will provide an introductory mini-course for the backgrounds, and then discuss four problems: constructions of self-complementary vertex-transitive graphs, self-complementary vertex-transitive graphs of order a product of two primes, self-complementary metacirculants, and self-complementary vertex-transitive graphs of prime-cube order. The main analysis on these problems relies on the two pivotal results due to Guralnick et al. [22] and Li, Praeger [31], which characterise the full automorphism group of a self-complementary vertex-transitive graph in the primitive and the imprimitive cases respectively.

For constructions of self-complementary vertex-transitive graphs, there are generally three known construction methods: construction by partitioning the complementing isomorphism orbits; construction using the coset graph; the lexicographic product. In this dissertation I shall develop various alternative construction methods. As a result, I find a family of self-complementary Cayley graphs of non-nilpotent groups, a new construction for self-complementary metacirculants of *p*-groups.

Original language | English |
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Qualification | Doctor of Philosophy |

Publication status | Unpublished - Sep 2014 |