Spectral Analysis of Growing Graphs: A Quantum Probability Point of View
This book is designed as a concise introduction to the recent achievements on spectral analysis of graphs or networks from the point of view of quantum (or non-commutative) probability theory. The main topics are spectral distributions of the adjacency matrices of finite or infinite graphs and their limit distributions for growing graphs. The main vehicle is quantum probability, an algebraic extension of the traditional probability theory, which provides a new framework for the analysis of adjacency matrices revealing their non-commutative nature. For example, the method of quantum decomposition makes it possible to study spectral distributions by means of interacting Fock spaces or equivalently by orthogonal polynomials. Various concepts of independence in quantum probability and corresponding central limit theorems are used for the asymptotic study of spectral distributions for product graphs.This book is written for researchers, teachers, and students interested in graph spectra, their (asymptotic) spectral distributions, and various ideas and methods on the basis of quantum probability. It is also useful for a quick introduction to quantum probability and for an analytic basis of orthogonal polynomials.
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A)xy adjacency algebra adjacency matrix algebraic probability space associated with Jacobi called Cartesian product characteristic polynomial coefficients a)n coincides complete graph connected graph converges convolution Corollary defined Definition deg(x denote density function det(x determinate moment problem distance-regular graph distinguished vertex distribution with parameter eigenvalue eigenvalue distribution Exercise finite graph Fock space associated given graph G Growing Graphs Hamming graph index set induced subgraph infinite type interacting Fock space isomorphic Jacobi coefficients a}n k-subalgebra Kronecker product Lemma Let G linear m-step walks matrix of G Mn(p monotone independent Obata obtain orthogonal polynomials permutation matrix Po(x probability measure probability measure pu product G1 Proof Let Proposition quantum decomposition random variable real random variable right-hand side Sect sequence Spectral Analysis star product Stieltjes transform Un(x unital k-algebra vacuum spectral distribution vector