Year: 2025
Type: article
Abstract: Let q be an odd power of a prime p , and $$S \subseteq \mathbb {F}_q^*$$ S ⊆ F q ∗ such that $$S=-S$$ S = - S and $$S/S \ne \mathbb {F}_q^*$$ S / S ≠ F q ∗ . We show that the clique number of the Cayley graph $$\operatorname {Cay}(\mathbb {F}_q^+,S)$$ Cay ( F q + , S ) is at most $$\sqrt{|S/S|}+\sqrt{q/p}$$ | S / S | + q / p , improving the best-known $$\sqrt{q}$$ q 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.
Source: Designs Codes and Cryptography
Author: Chi Hoi Yip
Institution: Georgia Institute of Technology
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
OpenAlex ID: W4413624477