On maximal cliques of Cayley graphs over fields

On eigenfunctions and maximal cliques of generalised Paley graphs of square order

“…Therefore, m | r 6 −1 r 2 −1 , and thus the subfield F r 2 forms a clique in GP (r 6 , m) = GP (q 2 , m). Note that ω(GP (q 2 , m)) = q = r 3 , so [30,Corollary 3.1] implies that the subfield F r 2 in fact forms a maximal clique (for otherwise ω(GP (r 6 , m)) ≥ r 4 ). In particular, for each…”

“…Inspired by Lemma 1.1 and Lemma 1.2, maximal cliques with subfield or subspace structure in Cayley graphs over fields are considered in [30]. For example, an explicit construction of maximal but not maximum cliques of size q 2 in cubic Paley graphs of order q 6 was given: the construction is simply the subfield with q 2 elements [30, Theorem 1.5].…”

On the clique number of Paley graphs of prime power order

“…We refer to [25, Section 1] and [22,Section 5.1] for the discussion on the best-known lower bound. Roughly speaking, the best-known lower bound is a polylogarithmic function in q, unless there is a subfield of F q which forms a clique in GP (q, d) (first observed by Broere, Döman, and Ridley [3], see also [24,26] for the maximality of such cliques). The latter case can be regarded as the exceptional case, in the sense that it is widely believed that the clique number of GP (q, d) is q o(1) (when q = p is a prime, this is implied by the Paley graph conjecture on double character sums [23,Section 2]) unless there is an obvious exceptionally large clique (that is, a subfield).…”

“…Observe that when q = p 3 and d ≥ 2 is a divisor of p 2 + p + 1, the subfield F p forms a clique in GP (q, d). In fact, in this case, it is known that F p forms a maximal clique in GP (q, d) [26], and it is tempting to conjecture that ω(GP (q, d)) = p since F p is the only "obvious" large clique. Indeed, when d = p 2 +p+1, Theorem 1.3 implies the sharp bound that ω(GP (q, d)) = p, although one can argue that in this case, the clique number is trivially p since it is a subfield cyclotomic graph.…”

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].…”