This is an introduction to representation theory and harmonic analysis on finite groups. This includes, in particular, Gelfand pairs (with applications to diffusion processes à la Diaconis) and induced representations (focusing on the little group method of Mackey and Wigner). We also discuss Laplace operators and spectral theory of finite regular graphs. In the last part, we present the representation theory of GL(2, Fq), the general linear group of invertible 2 × 2 matrices with coefficients in a finite field with q elements. More precisely, we revisit the classical Gelfand–Graev representation of GL(2, Fq) in terms of the so-called multiplicity-free triples and their associated Hecke algebras. The presentation is not fully self-contained: most of the basic and elementary facts are proved in detail, some others are left as exercises, while, for more advanced results with no proof, precise references are provided.

In this chapter we give a table with the feasible parameter sets of arbitrary strongly regular graphs on at most 512 vertices, and add comments about the known examples. These include: existence (and number of nonisomorphic examples), the parameters, the spectrum, information about being a descendant of a regular two-graph or being in the switching class of a regular two-graph, possible relation with a projective two-weight code,possible relation with a partial geometry, whether it is a conference graph or transversal design. Miscellaneous comments include references to earlier parts of the book, name of the graph, reason for non-existence, possible relation with a Steiner system, etc.