Exact values and improved bounds on the clique number of cyclotomic graphs

Abstract Let q be an odd power of a prime p , and $$S \subseteq \mathbb {F}_q^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>⊆</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:mrow> </mml:math> such that $$S=-S$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> and $$S/S \ne \mathbb {F}_q^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>/</mml:mo> <mml:mi>S</mml:mi> <mml:mo>≠</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:mrow> </mml:math> . We show that the clique number of the Cayley graph $$\operatorname {Cay}(\mathbb {F}_q^+,S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>Cay</mml:mo> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is at most $$\sqrt{|S/S|}+\sqrt{q/p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msqrt> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>S</mml:mi> <mml:mo>/</mml:mo> <mml:mi>S</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:msqrt> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>/</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msqrt> </mml:mrow> </mml:math> , improving the best-known $$\sqrt{q}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>q</mml:mi> </mml:msqrt> </mml:math> upper bound for many families of such graphs substantially. Such a new bound is strongest for cyclotomic graphs and in particular, it implies the first nontrivial upper bound on the clique number of all generalized Paley graphs of non-square order, extending the work of Hanson and Petridis. Moreover, our new bound is asymptotically sharp for an infinite family of generalized Paley graphs, and we further discover the first nontrivial family among them for which the clique number can be exactly determined. We also obtain a new lower bound on the number of directions determined by a large Cartesian product in the affine Galois plane AG (2, q ), which is sharp for infinite families.