SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS

Abstract

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, selfcomplementary vertex-transitive graphs of order a product of two primes, selfcomplementary 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. A complementing isomorphism of a self-complementary graph is an isomorphism between the graph and its complement. For the self-complementary vertex-transitive graphs whose automorphism groups are of affine type, we have obtained a characterisation of all their complementing isomorphisms. Furthermore, we provide a construction of self-complementary metacirculants which are Cayley graphs and have insoluble automorphism groups. This is the first known example with this property in the literature. For the self-complementary vertex-transitive graphs of order a product of two primes, we give a complete classification of these graphs: they are either a lexicographic product of two self-complementary vertex-transitive graphs of prime order, or a normal Cayley graph of an abelian group. A graph is called a metacirculant if its full automorphism group contains a transitive metacyclic subgroup. We shall explore self-complementary metacirculants and show that the full automorphism group of these graphs is either soluble or contains the only insoluble composition factor A5. This extends a result due to Li and Praeger [32] which says that the full automorphism group of a self-complementary circulant is soluble. Finally, we will investigate self-complementary vertex-transitive graphs of primecube order. We successfully show that for each type of the groups of prime-cube order, there exist self-complementary Cayley graphs of the corresponding groups. Moreover, we also gain a characterisation of all the self-complementary vertextransitive graphs of prime-cube order: they are normal Cayley graphs, or a lexicographic product of two smaller self-complementary vertex-transitive graphs, or their automorphism group is soluble.