








Zeta Functions of Graphs, A Stroll through the Garden












Publisher:  Cambridge University Press (Cambridge) 
  






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 LFunctions Edge Artin LFunctions  Path Artin LFunctions  NonIsomorphic 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 ErrorCorrecting Codes  Explicit Formulas  Again Chaos  Final Research Problems






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 numbertheoretic 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 nonisomorphic 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.












Artinized, Gauss sum, Frobenius automorphism, Galois theory, Graph, Kloosterman sum, Landau's theorem, Mobius function, Markov transition matrix, Orthogonality relations, Ramanujan, Theorem, Weil conjectures, Wigner semicircle distribution, Zeta functions







   
 
