Lecture 3.Possible analogues of the Hilton Milner theorem for Paley graphs of square order and Peis

In [7], a family of maximal but not maximum cliques in Paley graphs of square order was constructed. Moreover, it was conjectured that these cliques are second largest. In [9] a different family of maximal cliques with the proposed size we was constructed (the construction is based on an oval). However, recently it was shown in [10] that these two constructions are equivalent up to fractional transformation. Numerical experiments showed that when 25 ≤ q ≤ 83, these two families give rise to all maximal cliques with the proposed size (there are some extra cliques of the proposed size when 9 ≤ q ≤ 23). Thus, we propose the following stronger conjecture: if q ≥ 25, then each second largest maximal clique in the Paley graph P(q^2) is equivalent to a clique from the constructions proposed in [7] and [9].
In [8], it was noted that the fractional transformation from [10] can be described as a certain automorphism of the local subgraph of P(q^2).
In [11], it was noted that our conjecture is an analogue of the Hilton-Milner theorem.
In this lecture we discuss the details of this conjecture. We also discuss a similar conjecture for a special subclass of Peisert graphs.