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. Pitched at beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and diagrams, and exercises throughout, theoretical and computer-based.
  • ISBN13: 9780521113670
  • Publisher: Cambridge University Press
  • Pubilcation Year: 2010
  • Format: Hardcover
  • Pages: 00239
Specifications
FormatHardcover
SeriesCambridge Studies in Advanced Mathematics (Hardcover)
Series Volume Number128
Publication DateDecember 27, 2010

Zeta Functions of Graphs: A Stroll Through the Garden

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