Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions

TL;DR

This paper derives formulas for counting complete subgraphs in Peisert graphs using finite field hypergeometric functions, providing new proofs and asymptotic estimates for subgraphs of various sizes.

Contribution

It introduces a hypergeometric function approach to count subgraphs in Peisert graphs and offers new proofs and asymptotic formulas for these counts.

Findings

Formulas for complete subgraphs of order four in Peisert graphs.

New proof for counting triangles in Peisert graphs.

Asymptotic estimates for subgraphs of arbitrary size.

Abstract

For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle g^4\rangle$. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $P^\ast(q)$. We also give a new proof for the number of complete subgraphs of order three contained in $P^\ast(q)$ by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order $m$ in Peisert graphs.