Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Pitched at beginning graduate students, the book will also appeal to researchers. The book offers these students and researchers several examples of Mathematica illustrations and other content.
Riemann Zeta Function and other Zetas from Number Theory | Ihara Zeta Function | Selberg Zeta Function | Ruelle Zeta Function | Chaos | Ihara Zeta Function and the Graph Theory Prime Number Theorem | Ihara Zeta Function of a Weighted Graph | Regula Graphs, Location of Poles of the Ihara Zeta, Functional equations | Irregular graphs: What is the Riemann Hypothesis? | Discussion of Regular Ramanujan Graphs | Graph Theory prime Number Theorem | Edge and Path Zeta Functions | Edge Zeta Functions | Path Zeta Functions | Finite Unramified Galois Coverings of Connected Graphs | Finite Unramified Coverings and Galois Groups | Fundamental Theorem of Galois Theory | Behavior of Primes in Coverings | Frobenius Automorphisms | How to Construct Intermediate Coverings Using the Frobenius Automorphism | Artin L-Functions |Edge Artin L-Functions | Path Artin L-Functions | Non-Isomorphic Regular Graphs without Loops or Multiedges having the same Ihara Zeta Function | Chebotarev Density Theorem | Siegel Poles | Last Look at the Garden | An Application to Error-Correcting Codes | Explicit Formulas | Again Chaos | Final Research Problems
, Applied Mathematics
, Calculus and Analysis