Zeta functions of graphs: a stroll through the garden
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2011
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge studies in advanced mathematics
128 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020690832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XII, 239 S. Ill., graph. Darst. |
| ISBN: | 9780521113670 |
Internformat
MARC
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| 300 | |a XII, 239 S. |b Ill., graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1825578095351955456 |
|---|---|
| adam_text |
Contents
List of illustrations page
viii
Preface
xi
Part I A quick look at various
zeta
functions
1
1
Riemann
zeta
function and other
zetas
from
number theory
3
Ihara
zeta
function
10
2.1
The usual hypotheses and some definitions
10
2.2
Primes in X
11
2.3
Ihara
zeta
function
12
2.4
Fundamental group of a graph and its connection
with primes
13
2.5
Ihara determinant formula
17
2.6
Covering graphs
20
2.7
Graph theory prime number theorem
21
3
Selberg
zeta
function
22
4
Ruelle
zeta
function
27
5 Chaos
31
Part II Ihara
zeta
function and the graph theory prime
number theorem
43
6
Ihara
zeta
function of a weighted graph
45
vi
Contents
7
Regular
graphs, location of poles of the Ihara
zeta,
functional equations
47
8
Irregular graphs: what is the Riemann hypothesis?
52
9
Discussion of regular Ramanujan graphs
61
9.1
Random walks on regular graphs
61
9.2
Examples: the Paley graph, two-dimensional Euclidean
graphs, and the graphs of Lubotzky, Phillips, and
Sárnak
63
9.3
Why the Ramanujan bound is best possible
(Alon and Boppana theorem)
68
9.4
Why are Ramanujan graphs good expanders?
70
9.5
Why do Ramanujan graphs have small diameters?
73
10
Graph theory prime number theorem
75
10.1
Which graph properties are determined by the Ihara
zeta?
78
Partili
Edge and path
zeta
functions
81
11
Edge
zeta
functions
83
11.1
Definitions and Bass's proof of the Ihara three-term
determinant formula
83
11.2
Properties of W\ and a proof of the theorem of Kotani
and Sunada
90
12
Path
zeta
functions
98
Part IV Finite unramified Galois coverings of connected
graphs
103
13
Finite unramified coverings and Galois groups
105
13.1
Definitions
105
13.2
Examples of coverings 111
13.3
Some ramification experiments
115
14
Fundamental theorem of Galois theory
117
15
Behavior of primes in coverings
128
16
Frobenius automorphisms
133
17
How to construct intermediate coverings using the
Frobenius automorphism
141
Contents
vii
18 Artin
¿-functions
144
18.1 Brief
survey on representations of finite groups
144
18.2
Definition of the Artin-Ihara L-function
148
18.3
Properties of Artin-Ihara
L
-functions
154
18.4
Examples of factorizations of Artin-Ihara
L
-functions
157
19
Edge
Artin L-fimctions 164
19.1
Definition and properties of edge
Artin L
-functions
164
19.2
Proofs of determinant formulas for edge
Artin L
-functions
169
19.3
Proof of the induction property
173
20
Path Artin L-functions
178
20.1
Definition and properties of path Artin L-functions
178
20.2
Induction property
180
21
Non-isomorphic regular graphs without loops or multiedges
having the same Ihara
zeta
function
186
22
Chebotarev density theorem
194
23 Siegel
poles
200
23.1
Summary of
Siegel
pole results
200
23.2
Proof of Theorems
23.3
and
23.5 202
23.3
General case; inflation and deflation
206
Part V Last look at the garden
209
24
An application to error-correcting codes
211
25
Explicit formulas
216
26
Again chaos
218
27
Final research problems
227
References
230
Index
236 |
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| author | Terras, Audrey 1942- |
| author_GND | (DE-588)115217320 |
| author_facet | Terras, Audrey 1942- |
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| bvnumber | BV036774039 |
| classification_rvk | SK 180 |
| ctrlnum | (OCoLC)698487595 (DE-599)BVBBV036774039 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV036774039 |
| illustrated | Illustrated |
| indexdate | 2025-03-03T13:02:21Z |
| institution | BVB |
| isbn | 9780521113670 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020690832 |
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| physical | XII, 239 S. Ill., graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Terras, Audrey 1942- Verfasser (DE-588)115217320 aut Zeta functions of graphs a stroll through the garden Audrey Terras 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2011 XII, 239 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 128 Zetafunktion (DE-588)4190764-4 gnd rswk-swf Zetafunktion (DE-588)4190764-4 s DE-604 Cambridge studies in advanced mathematics 128 (DE-604)BV000003678 128 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020690832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Terras, Audrey 1942- Zeta functions of graphs a stroll through the garden Cambridge studies in advanced mathematics Zetafunktion (DE-588)4190764-4 gnd |
| subject_GND | (DE-588)4190764-4 |
| title | Zeta functions of graphs a stroll through the garden |
| title_auth | Zeta functions of graphs a stroll through the garden |
| title_exact_search | Zeta functions of graphs a stroll through the garden |
| title_full | Zeta functions of graphs a stroll through the garden Audrey Terras |
| title_fullStr | Zeta functions of graphs a stroll through the garden Audrey Terras |
| title_full_unstemmed | Zeta functions of graphs a stroll through the garden Audrey Terras |
| title_short | Zeta functions of graphs |
| title_sort | zeta functions of graphs a stroll through the garden |
| title_sub | a stroll through the garden |
| topic | Zetafunktion (DE-588)4190764-4 gnd |
| topic_facet | Zetafunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020690832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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