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Zeta Functions of Graphs
A Stroll through the Garden


Part of Cambridge Studies in Advanced Mathematics

  • Date Published: November 2010
  • availability: Available
  • format: Hardback
  • isbn: 9780521113670


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About the Authors
  • 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.

    • Makes connections with quantum chaos and random matrix theory, plus Ramanujan graphs, which are of interest to computer scientists
    • Explains key ideas using lots of well-chosen illustrations, alongside theoretical and computer-based exercises
    • Perfect for beginning graduate students, or established researchers who want a stimulating introduction to the topic
    Read more

    Reviews & endorsements

    'The book is very appealing through its informal style and the variety of topics covered and may be considered the standard reference book in this field.' Zentralblatt MATH

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    Product details

    • Date Published: November 2010
    • format: Hardback
    • isbn: 9780521113670
    • length: 252 pages
    • dimensions: 235 x 157 x 19 mm
    • weight: 0.53kg
    • contains: 65 b/w illus. 11 colour illus. 95 exercises
    • availability: Available
  • Table of Contents

    List of illustrations
    Part I. A Quick Look at Various Zeta Functions:
    1. Riemann's zeta function and other zetas from number theory
    2. Ihara's zeta function
    3. Selberg's zeta function
    4. Ruelle's zeta function
    5. Chaos
    Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem:
    6. Ihara zeta function of a weighted graph
    7. Regular graphs, location of poles of zeta, functional equations
    8. Irregular graphs: what is the RH?
    9. Discussion of regular Ramanujan graphs
    10. The graph theory prime number theorem
    Part III. Edge and Path Zeta Functions:
    11. The edge zeta function
    12. Path zeta functions
    Part IV. Finite Unramified Galois Coverings of Connected Graphs:
    13. Finite unramified coverings and Galois groups
    14. Fundamental theorem of Galois theory
    15. Behavior of primes in coverings
    16. Frobenius automorphisms
    17. How to construct intermediate coverings using the Frobenius automorphism
    18. Artin L-functions
    19. Edge Artin L-functions
    20. Path Artin L-functions
    21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function
    22. The Chebotarev Density Theorem
    23. Siegel poles
    Part V. Last Look at the Garden:
    24. An application to error-correcting codes
    25. Explicit formulas
    26. Again chaos
    27. Final research problems

  • Author

    Audrey Terras, University of California, San Diego
    Audrey Terras is Professor of Mathematics at the University of California, San Diego.

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