# Graphs and Groups, Geometries and GAP (G2G2) Summer School - External Satellite Conference of 8ECM

27 June 2021 to 3 July 2021
Rogla, Slovenia
UTC timezone

## On maximal cliques in Paley graphs of square order

Not scheduled
20m
Rogla, Slovenia

#### Rogla, Slovenia

Hotel Planja, Rogla 1, 3214 Zreče, Slovenia
Oral presentation

### Speaker

Dr Sergey Goryainov (Chelyabinsk State University)

### Description

In [1], Blokhuis studied maximum cliques in Paley graphs of square order $𝑃(𝑞^2)$. It was shown that a clique of size $q$ in $𝑃(𝑞^2)$ is necessarily a quadratic line in the corresponding affine plane $𝐴(2,𝑞)$.

Let $𝑟(𝑞)$ denote the reminder after division of $𝑞$ by $4$. In [2], for any odd prime power $𝑞$, a maximal (but not maximum) clique in $𝑃(𝑞^2)$ of size $\frac{𝑞+𝑟(𝑞)}{2}$ was constructed.

In [3], for any odd prime power $𝑞$, a maximal clique in $𝑃(𝑞^2)$ of the same size $\frac{𝑞+𝑟(𝑞)}{2}$ was constructed. This clique was shown to have a remarkable connection with eigenfunctions of $𝑃(𝑞^2)$ that have minimum cardinality of support $𝑞+1$.

In this talk, we discuss the constructions of maximal cliques from [2] and [3] and establish a correspondence between them.

Acknowledgments. Sergey Goryainov and Leonid Shalaginov are supported by RFBR according to the research project 20-51-53023.

References
[1] A. Blokhuis, On subsets of $𝐺𝐹(𝑞^2)$ with square differences. Indag. Math. 46 (1984) 369–372.
[2] R. D. Baker, G. L. Ebert, J. Hemmeter, A. J. Woldar, Maximal cliques in the Paley graph of square order. J. Statist. Plann. Inference 56 (1996) 33–38.
[3] S. V. Goryainov, V. V. Kabanov, L. V. Shalaginov, A. A. Valyuzhenich, On eigenfunctions and maximal cliques of Paley graphs of square order. Finite Fields and Their Applications 52 (2018) 361–369.
[4] S. V. Goryainov, A. V. Masley, L. V. Shalaginov, On a correspondence between maximal cliques in Paley graphs of square order. arXiv:2102.03822 (2021).

### Primary author

Dr Sergey Goryainov (Chelyabinsk State University)