Zeta Functions of Graphs: A Stroll through the GardenGraph 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. |
From inside the book
Results 1-5 of 18
Page i
... zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann or Dedekind zeta functions. For example, there is a Riemann hypothesis ...
... zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann or Dedekind zeta functions. For example, there is a Riemann hypothesis ...
Page xi
... Dedekind zeta function of an algebraic number field and its analog for function fields . Many Riemann hypotheses have been formulated and a few proved . The statistics of the complex zeros of zeta have been connected with the statistics ...
... Dedekind zeta function of an algebraic number field and its analog for function fields . Many Riemann hypotheses have been formulated and a few proved . The statistics of the complex zeros of zeta have been connected with the statistics ...
Page xii
... Dedekind zeta functions of Galois extensions of number fields. Most of this book arises from joint work with Harold Stark. Thanks are due to the many people who listened to my lectures on this book and helped with the research, Matthew ...
... Dedekind zeta functions of Galois extensions of number fields. Most of this book arises from joint work with Harold Stark. Thanks are due to the many people who listened to my lectures on this book and helped with the research, Matthew ...
Page 3
... Dedekind, Dirichlet, Hecke, Takagi, Artin, and others. Here we will concentrate on the original, namely Riemann's zeta function. The definition is as follows. Riemann's zeta function for s∈ C with Re s > 1 is defined to be ζ(s) ...
... Dedekind, Dirichlet, Hecke, Takagi, Artin, and others. Here we will concentrate on the original, namely Riemann's zeta function. The definition is as follows. Riemann's zeta function for s∈ C with Re s > 1 is defined to be ζ(s) ...
Page 4
... zeta zeros " : http://www.maa.org/editorial/mathgames You win $ 1 million if you have a proof of the Riemann ... Dedekind zeta function of an algebraic number field K , such as K = Q ( √2 ) , for example . This zeta is an ...
... zeta zeros " : http://www.maa.org/editorial/mathgames You win $ 1 million if you have a proof of the Riemann ... Dedekind zeta function of an algebraic number field K , such as K = Q ( √2 ) , for example . This zeta is an ...
Contents
II Ihara zeta function and the graph theory prime number theorem | 43 |
III Edge and path zeta functions | 81 |
IV Finite unramified Galois coverings of connected graphs | 103 |
V Last look at the garden | 209 |
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Common terms and phrases
adjacency matrix algebraic number analog Artin L-functions backtrackless base graph Cayley graph circle conjecture conjugacy class corresponding cosets covering graphs cube cycle Dedekind zeta function defined Definition density det(I determinant formula directed edge edge adjacency matrix edge L-function edge zeta function eigenvalues entries example Exercise Figure follows Frobenius automorphism fundamental group Galois group Galois theory graph coverings graph theory group G H₁ Hoory Ihara zeta function intermediate cover intermediate graphs intermediate to Y/X irreducible isomorphic isospectral k-regular LE(W Lemma length level spacings lift loops Lubotzky Math normal cover normalized Frobenius automorphism number fields number theory permutation prime number theorem proof Proposition prove quadratic cover radius Ramanujan graphs random regular graphs Riemann hypothesis Sarnak Selberg sheet Siegel pole spanning tree spectrum Terras tetrahedron unramified usual hypotheses vector W₁ Y/X is normal zeros