Zeta Functions of Graphs: A Stroll through the Garden (Cambridge Studies in Advanced Mathematics, Series Number 128)

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-- Machine Generated Contents Note: List Of Illustrations; Preface; 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; References; Index. Audrey Terras. Includes Bibliographical References (p. 230-235) And Index. Cover......Page 1 Half-title......Page 3 Series-title......Page 4 Title......Page 5 Copyright......Page 6 Contents......Page 7 Illustrations......Page 10 Preface......Page 13 I A quick look at various zeta functions......Page 15 1 Riemann zeta function and other zetas from number theory......Page 17 2.1 The usual hypotheses and some definitions......Page 24 2.2 Primes in X......Page 25 2.3 Ihara zeta function......Page 26 2.4 Fundamental group of a graph and its connection with primes......Page 27 2.5 Ihara determinant formula......Page 31 2.6 Covering graphs......Page 34 2.7 Graph theory prime number theorem......Page 35 3 Selberg zeta function......Page 36 4 Ruelle zeta function......Page 41 5 Chaos......Page 45 II Ihara zeta function and the graph theory prime number theorem......Page 57 6 Ihara zeta function of a weighted graph......Page 59 7 Regular graphs, location of poles of the Ihara zeta, functional equations......Page 61 8 Irregular graphs: what is the Riemann hypothesis?......Page 66 9.1 Random walks on regular graphs......Page 75 9.2 Examples: the Paley graph, two-dimensional Euclidean graphs, and the graphs of Lubotzky, Phillips, and Sarnak......Page 77 9.3 Why the Ramanujan bound is best possible (Alon and Boppana theorem)......Page 82 9.4 Why are Ramanujan graphs good expanders?......Page 84 9.5 Why do Ramanujan graphs have small diameters?......Page 87 10 Graph theory prime number theorem......Page 89 10.1 Which graph properties are determined by the Ihara zeta?......Page 92 III Edge and path zeta functions......Page 95 11.1 Definitions and Bass's proof of the Ihara three-term determinant formula......Page 97 11.2 Properties of W1 and a proof of the theorem of Kotaniand Sunada......Page 104 12 Path zeta functions......Page 112 IV Finite unramified Galois coverings of connected graphs......Page 117 13.1 Definitions......Page 119 13.2 Examples of coverings......Page 125 13.3 Some ramification experiments......Page 129 14 Fundamental theorem of Galois theory......Page 131 15 Behavior of primes in coverings......Page 142 16 Frobenius automorphisms......Page 147 17 How to construct intermediate coverings using the Frobenius automorphism......Page 155 18.1 Brief survey on representations of finite groups......Page 158 18.2 Definition of the Artin–Ihara L-function......Page 162 18.3 Properties of Artin–Ihara L-functions......Page 168 18.4 Examples of factorizations of Artin–Ihara L-functions......Page 171 19.1 Definition and properties of edge Artin L-functions......Page 178 19.2 Proofs of determinant formulas for edge Artin L-functions......Page 183 19.2.1 Bass proof of Ihara theorem for Artin L-functions......Page 184 19.3 Proof of the induction property......Page 187 20.1 Definition and properties of path Artin L-functions......Page 192 20.2 Induction property......Page 194 21 Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function......Page 200 22 Chebotarev density theorem......Page 208 23.1 Summary of Siegel pole results......Page 214 23.2 Proof of Theorems 23.3 and 23.5......Page 216 23.3 General case inflation and deflation......Page 220 V Last look at the garden......Page 223 24 An application to error-correcting codes......Page 225 25 Explicit formulas......Page 230 26 Again chaos......Page 232 27 Final research problems......Page 241 References......Page 244 Index......Page 250 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.