Circulant Graphs: Symmetry, Paley Graphs, and Ramsey Bounds

Abstract

Circulant graphs provide a natural meeting point for algebra, number theory, and graph theory. We begin by viewing them as Cayley graphs of cyclic groups, which makes their vertex symmetry and connectivity transparent. We then characterise circulant graphs through their automorphism groups, introduce Paley graphs through quadratic residues, and prove that Paley graphs are connected, arc-transitive, and self-complementary. Finally, we use these symmetries to verify explicit circulant constructions giving lower bounds for small Ramsey numbers.