Spectral Theory of Cayley Graphs of Finite Groups

Abstract

This thesis explores the spectral properties of Cayley graphs and their connections to representation theory. Spectral graph theory studies the eigenvalues of adjacency and Laplacian matrices, which reveal structural properties of graphs. When a group acts transitively on a graph, its adjacency and Laplacian spectra are closely related and can often be analyzed through group representations. The Cayley graph of a group, defined with respect to a generating set, provides a natural framework for studying spectral properties using character theory. Key results on Markov chain theory by J. R. Norris and those of Diaconis, Bayer, and Aldous on card shuffling are examined in this context. Additionally, Lov´asz’s work on the eigenvalues of graphs in terms of character theory is discussed. The thesis concludes with explicit calculations of the spectra of the Cayley graphs of dihedral groups and S 4 , using these theoretical insights.