| Description |
1 online resource (xii, 239 pages) : illustrations (1 color). |
| Physical Medium |
polychrome |
| Description |
text file |
| Series |
Cambridge studies in advanced mathematics ; 128
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Cambridge studies in advanced mathematics ; 128.
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| Summary |
"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"-- Provided by publisher. |
| Bibliography |
Includes bibliographical references (pages 230-235) and index. |
| Contents |
A quick look at various zeta functions. Riemann's zeta function and other zetas from number theory -- Ihara's zeta function -- Selberg's zeta function -- Ruelle's zeta function -- Chaos -- Ihara's zeta function and the graph theory prime number theorem. Ihara zeta function of a weighted graph -- Regular graphs, location of poles of zeta, functional equations -- Irregular graphs: what is the RH? -- Discussion of regular Ramanujan graphs -- The graph theory prime number theorem --Edge and path zeta functions. The edge zeta function -- Path zeta functions -- Finite unramified Galois coverings of connected graphs. Finite unramified coverings and Galois groups -- Fundamental theorem of Galois theory -- Behavior of primes in coverings -- Frobenius automorphisms -- How to construct intermediate coverings using the Frobenius automorphism -- Artin L-functions -- Edge Artin L-functions -- Path Artin L-functions -- Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function -- The Chebotarev density theorem -- Siegel poles -- Last look at the garden. An application to error-correcting codes -- Explicit formulas -- Again chaos -- Final research problems. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Graph theory.
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Graph theory. |
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Functions, Zeta.
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Functions, Zeta. |
| Genre/Form |
Electronic books.
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Electronic book.
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| Other Form: |
Print version: Terras, Audrey. Zeta functions of graphs. New York : Cambridge University Press, 2010 9780521113670 (DLC) 2010024611 (OCoLC)639166318 |
| ISBN |
9780511918681 (electronic book) |
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0511918682 (electronic book) |
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9780511914911 (electronic book) |
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0511914911 (electronic book) |
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9780511760426 (electronic book) |
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0511760426 (electronic book) |
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9780521113670 (hardback) |
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0521113679 (hardback) |
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