On the Eigenvalue Spectrum of Cayley Graphs: Connections to Group Structure and Expander Properties

Cayley graphs sit at the intersection of algebra, geometry, and theoretical computer science. Their spectra encode fine structural information about both the underlying group and the graph itself. Building on classical work of Alon–Milman, Dodziuk, Margulis, Lubotzky–Phillips–Sarnak, and many others, we develop a unified representation-theoretic framework that yields several new results. We establish a monotonicity principle showing that the algebraic connectivity never decreases when generators are added. We provide closed-form spectra for canonical 3-regular dihedral Cayley graphs, with exact spectral gaps. We prove a quantitative obstruction demonstrating that bounded-degree Cayley graphs of groups with growing abelian quotients cannot form expander families. In addition, we present two universal comparison theorems: one for quotients and one for direct products of groups. We also derive explicit eigenvalue formulas for class-sum-generating sets together with a Hoffman-type second-moment bound for all Cayley graphs. We also establish an exact relation between the Laplacian spectra of a Cayley graph and its complement, giving a closed-form expression for the complementary spectral gap. These results give new tools for deciding when a given family of Cayley graphs can or cannot expand, sharpening and extending several classical criteria.