A Brief Introduction to Spectral Graph Theory

Spectral graph theory starts by associating matrices to graphs – notably, the adjacency matrix and the Laplacian matrix. The general theme is then, firstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to structural properties of graphs. As it turns out, the spectral perspective is a powerful tool. Some of its loveliest applications concern facts that are, in principle, purely graph theoretic or combinatorial. This text is an introduction to spectral graph theory, but it could also be seen as an invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and how they come together. This part provides an appealing motivation and context of the second, spectral, half. The text is enriched by many exercises and their solutions. The target audience are students from the upper undergraduate level onwards. We assume only a familiarity with linear algebra and basic group theory. Graph theory, finite fields, and character theory for abelian groups receive a concise overview and render the text essentially self-contained. Keywords: Adjacency eigenvalues of graphs, Laplacian eigenvalues of graphs, Cayley graphs, algebraic graphs over finite fields, character sums