Paley-like quasi-random graphs arising from polynomials

TL;DR

This paper introduces new polynomial-based constructions of quasi-random graphs resembling Paley graphs, expanding understanding of their properties and providing bounds on their clique and independence numbers.

Contribution

It presents novel non-Cayley quasi-random graphs from polynomials over finite fields, unifying various graph classes and analyzing their combinatorial properties.

Findings

New polynomial-based quasi-random graphs similar to Paley graphs.

Lower bounds established for clique and independence numbers.

Unified framework for graphs from polynomials over finite fields.

Abstract

Paley graphs and Paley sum graphs are classical examples of quasi-random graphs. In this paper, we provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs arising from polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs arising from Diophantine tuples and their generalizations. We also obtain lower bounds on the clique and independence numbers of the graphs in these families.