Strongly regular graphs, association schemes and Gauss sums

Abstract

Gauss sums play an important role in the construction of strongly regular Cayley graphs and association schemes. Compared with other approaches to the constructions of strongly regular graphs, the method using Gauss sums requires a lot of background knowledge from algebra and number theory. In [50], Schmidt and White provided a conjecture on cyclotomic strongly regular graphs which contains 11 sporadic examples of cyclotomic strongly regular graphs. We rst generalize one of their sporadic examples to an innite family of strongly regular graphs by using a union of cyclotomic classes. We do so by rst deriving expressions for the (restricted) eigenvalues of Cayley graphs without evaluating Gauss sums explicitly, and then giving conditions that determined candidate Cayley graphs be strongly regular. A. V. Ivanov's conjecture on amorphic association schemes was rst disproved by Van Dam. Before the work of this dissertation, only nitely many counterexamples were known. In the dissertation, we shall give 15 innite families of counter examples to Ivanov's conjecture. Moreover, our families of association schemes are pseudocyclic. We shall prove this fact by using the properties of Gauss sums of the index 2.