Paley-like quasi-random graphs arising from polynomials

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 coming from Diophantine tuples and their generalizations. We also provide new lower bounds on the clique number and independence number of general quasi-random graphs. In particular, we give a sufficient condition for the clique number of quasi-random graphs of n vertices to be at least (1-o (1) ₃. ₀₀₈n. Such a condition applies to many classical quasi-random graphs, including Paley graphs and Paley sum graphs, as well as some new Paley-like graphs we construct.