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

Abstract

Let GP$(q^2,m)$ be the $m$-Paley graph defined on the finite field with order $q^2$. We study eigenfunctions and maximal cliques in generalised Paley graphs GP$(q^2,m)$, where $m\mid (q+1)$. In particular, we explicitly construct maximal cliques of size $\frac{q+1}{m}$ or $\frac{q+1}{m}+1$ in GP$(q^2,m)$, and show the weight-distribution bound on the cardinality of the support of an eigenfunction is tight for the smallest eigenvalue $-\frac{q+1}{m}$ of GP$(q^2,m)$. These new results extend the work of Baker et. al and Goryainov et al. on Paley graphs of square order. We also study the stability of the Erdős-Ko-Rado theorem for GP$(q^2,m)$ (first proved by Sziklai).

References in corpus (11)

• Refined Estimates Concerning Sumsets Contained in the Roots of Unity
• To the theory of $q$-ary Steiner and other-type trades
• On the clique number of Paley graphs of prime power order
• On the directions determined by Cartesian products and the clique number of generalized Paley graphs
• Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields
• Maximal integral point sets in affine planes over finite fields
• Maximal integral point sets in affine planes over finite fields
• On maximal cliques of Cayley graphs over fields
• On eigenfunctions and maximal cliques of Paley graphs of square order
• The EKR-module property of pseudo-Paley graphs of square order
• On a correspondence between maximal cliques in Paley graphs of square order

Cited by in corpus (2)

• Erdős-Ko-Rado theorem in Peisert-type graphs
• Extremal Peisert-type graphs without the strict-EKR property