Mathematical essay
Circulant Graphs: Symmetry, Paley Graphs, and Ramsey Bounds
My third-year undergraduate expository essay connecting circulant graphs with group theory, number theory, Paley graphs, and Ramsey bounds.
19 pages Revised in 2025
Abstract
Circulant graphs provide a natural meeting point for algebra, number theory, and graph theory. This essay views them as Cayley graphs of cyclic groups, examines the Cayley isomorphism problem, and introduces Paley graphs through quadratic residues. It proves that Paley graphs are connected, arc-transitive, and self-complementary, then shows how the symmetry of circulant and Paley graphs can be used to construct lower bounds for Ramsey numbers.