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Zeta Functions of Graphs

Book Description

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. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. List of illustrations
  6. Preface
  7. Part I: A quick look at various zeta functions
    1. 1. Riemann zeta function and other zetas from number theory
    2. 2. Ihara zeta function
      1. 2.1 The usual hypotheses and some definitions
      2. 2.2 Primes in X
      3. 2.3 Ihara zeta function
      4. 2.4 Fundamental group of a graph and its connection with primes
      5. 2.5 Ihara determinant formula
      6. 2.6 Covering graphs
      7. 2.7 Graph theory prime number theorem
    3. 3. Selberg zeta function
    4. 4. Ruelle zeta function
    5. 5. Chaos
  8. Part II: Ihara zeta function and the graph theory prime number theorem
    1. 6. Ihara zeta function of a weighted graph
    2. 7. Regular graphs, location of poles of the Ihara zeta, functional equations
    3. 8. Irregular graphs: what is the Riemann hypothesis?
    4. 9. Discussion of regular Ramanujan graphs
      1. 9.1 Random walks on regular graphs
      2. 9.2 Examples: the Paley graph, two-dimensional Euclidean graphs, and the graphs of Lubotzky, Phillips, and Sarnak
      3. 9.3 Why the Ramanujan bound is best possible (Alon and Boppana theorem)
      4. 9.4 Why are Ramanujan graphs good expanders?
      5. 9.5 Why do Ramanujan graphs have small diameters?
    5. 10. Graph theory prime number theorem
      1. 10.1 Which graph properties are determined by the Ihara zeta?
  9. Part III: Edge and path zeta functions
    1. 11. Edge zeta functions
      1. 11.1 Definitions and Bass’s proof of the Ihara three-term determinant formula
      2. 11.2 Properties of W1 and a proof of the theorem of Kotani and Sunada
    2. 12. Path zeta functions
  10. Part IV: Finite unramified Galois coverings of connected graphs
    1. 13. Finite unramified coverings and Galois groups
      1. 13.1 Definitions
      2. 13.2 Examples of coverings
      3. 13.3 Some ramification experiments
    2. 14. Fundamental theorem of Galois theory
    3. 15. Behavior of primes in coverings
    4. 16. Frobenius automorphisms
    5. 17. How to construct intermediate coverings using the Frobenius automorphism
    6. 18. Artin L-functions
      1. 18.1 Brief survey on representations of finite groups
      2. 18.2 Definition of the Artin–Ihara L-function
      3. 18.3 Properties of Artin–Ihara L-functions
      4. 18.4 Examples of factorizations of Artin–Ihara L-functions
    7. 19. Edge Artin L-functions
      1. 19.1 Definition and properties of edge Artin L-functions
      2. 19.2 Proofs of determinant formulas for edge Artin L-functions
      3. 19.3 Proof of the induction property
    8. 20. Path Artin L-functions
      1. 20.1 Definition and properties of path Artin L-functions
      2. 20.2 Induction property
    9. 21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function
    10. 22. Chebotarev density theorem
    11. 23. Siegel poles
      1. 23.1 Summary of Siegel pole results
      2. 23.2 Proof of Theorems 23.3 and 23.5
      3. 23.3 General case; inflation and deflation
  11. Part V: Last look at the garden
    1. 24. An application to error-correcting codes
    2. 25. Explicit formulas
    3. 26. Again chaos
    4. 27. Final research problems
  12. References
  13. Index