Matrix inequalities for iterative systems (Q5891012)

One of the first things learned in algebraic graph theory is that each entry of the \(k\)-th power of the adjacency matrix of a graph equals the number of walks of length \(k\) between the vertex corresponding to the row and the vertex corresponding to the column. Also, the number of closed walks of length \(k\) equals the sum of the \(k\)-th powers of the eigenvalues of the adjacency matrix. These beautiful relations between combinatorial parameters (walks) and algebraic parameters (matrix powers, eigenvalues) are useful in many instances and belong to a larger class of connections between graph walks, matrix powers and eigenvalues.NEWLINENEWLINEThis book is based on the author's habilitation thesis and gives a unified presentation of various inequalities for the number of walks in graphs and for the sum of entries of matrix powers. The main goal of the book is to give a clear overview of common principles underlying the huge number of different results. The book consists of 165 pages (plus index and a 30+ pages bibliography) which are grouped into 4 main groups: introduction (which includes a section on motivation), undirected graphs/Hermitian matrices (which takes the bulk of the book, 50+ pages), directed graphs/nonsymmetric matrices and Applications. The motivation section describes connections to other areas such as random walks, automata and formal languages, population genetics, statistical mechanics and theoretical chemistry.NEWLINENEWLINEThis is a nice and well-written book that will be useful to researchers interested in the connections between graph walks, matrix powers and eigenvalues.