The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes

Abstract

Blokhuis showed that Paley graphs with square order have the Erdős-Ko-Rado (EKR) property in the sense that all maximum cliques are canonical. In our previous work, we extended the EKR property of Paley graphs to certain Peisert graphs and generalized Peisert graphs. In this paper, we propose a conjecture which generalizes the EKR property of Paley graphs, and can be viewed as an analogue of Chvátal's Conjecture for families of set systems. As a partial progress, we prove that maximum cliques in pseudo-Paley graphs have a rigid structure.

The EKR-Module Property of Pseudo-Paley Graphs of Square Order

“…Theorem 1 was first proved by Blokhuis [7]. Extensions and generalizations of Theorem 1 can be found in [13,30,25,3,4]. A Fourier analytic approach was recently proposed in [34,Section 4.4].…”

The EKR-module property of pseudo-Paley graphs of square order

“…The following theorem characterizes the maximum cliques in P q 2 . can be found in [BF91,Szi99,Mul09,AY21a,AY21b]. A Fourier analytic approach was recently proposed in [Yip21,Section 4.4].…”

“…We remark that Peisert-type graphs in fact form a subfamily of a well-known family of strongly regular Cayley graphs defined on finite fields due to Brouwer, Wilson, and Xiang [BWX99]: the connection set is a union of semiprimitive cyclotomic classes of F q 2 . However, their proof heavily relied on the fact we can compute semi-primitive Gauss sums explicitly using Stickelberger's theorem and its variants; see [BWX99, Proposition 1] and [AY21a,Corollary 3.6]. We will see that Theorem 1.4 can be proved using a purely combinatorial argument, thus giving an elementary proof of the corollary below.…”

Hypergeometric functions for Dirichlet characters and Peisert-like graphs

“…. Various properties of Peisert graphs have been studied, for example, their automorphism groups by Peisert himself in [17], pseudo-random properties in [11], maximal and maximum cliques in [20] and [3], critical groups of the graphs in [18], etc. Peisert graphs have been used to produce binary and ternary codes from their adjacency matrices in [10].…”

“…Besides Paley graphs being generalized (see for example [4,12]), Peisert graphs have also been generalized into graphs called generalized Peisert or Peisert type graphs and their cliques have been studied in [2,3,14]. In this article, we introduce a Peisert-like graph on the commutative ring Z n , for suitable n. Computing the number of cliques in Paley, Peisert and Paley-type graphs has been of interest, for instance see [1,4,5,7].…”