Eigenvalues of the Laplacian matrix of a graph have been widely used in studying connectivity and expansion properties of networks, and also in analyzing random walks on a graph. Independently, statisticians introduced various optimality criteria in experimental design, the goal being to obtain more accurate estimates of quantities of interest in an experiment. It turns out that the most popular of these optimality criteria for block designs are determined by the Laplacian eigenvalues of the concurrence graph, or of the Levi graph, of the design. The most important optimality criteria, called A (average), D (determinant) and E (extreme), are related to the conductance of the graph as an electrical network, the number of spanning trees, and the isoperimetric properties of the graphs, respectively. The number of spanning trees is also an evaluation of the Tutte polynomial of the graph, and is the subject of the Merino–Welsh conjecture relating it to acyclic and totally cyclic orientations, of interest in their own right. This chapter ties these ideas together, building on the work in [4] and [5].

We give an introduction to a topic in the “stable algebra of matrices,” as related to certain problems in symbolic dynamics. We introduce enough symbolic dynamics to explain these connections, but the algebra is of independent interest and can be followed with little attention to the symbolic dynamics. This “stable algebra of matrices” involves the study of properties and relations of square matrices over a semiring S, which are invariant under two fundamental equivalence relations: shift equivalence and strong shift equivalence. When S is a field, these relations are the same, and matrices over S are shift equivalent if and only if the nonnilpotent parts of their canonical forms are similar. We give a detailed account of these relations over other rings and semirings. When S is a ring, this involves module theory and algebraic K theory. We discuss in detail and contrast the problems of characterizing the possible spectra, and the possible nonzero spectra, of nonnegative real matrices.We also review key features of the automorphism group of a shift of finite type; the recently introduced stabilized automorphism group; and the work of Kim, Roush and Wagoner giving counterexamples to Williams’ shift equivalence conjecture.

These lecture notes provide quantum probabilistic concepts and methods for spectral analysis of graphs, in particular, for the study of asymptotic behavior of the spectral distributions of growing graphs. Quantum probability theory is an algebraic generalization of classical (Kolmogorovian) probability theory, where an element of a (not necessarily commutative) ∗-algebra is treated as a random variable. In this aspect the concepts and methods peculiar to quantum probability are applied to the spectral analysis of adjacency matrices of graphs. In particular, we focus on the method of quantum decomposition and the use of various concepts of independence. The former discloses the noncommutative nature of adjacency matrices and gives a systematic method of computing spectral distributions. The latter is related to various graph products and provides a unified aspect in obtaining the limit spectral distributions as corollaries of various central limit theorems.

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.