Self-complementary vertex-transitive graphs

[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.