Book contents
- Frontmatter
- Contents
- List of illustrations
- Preface
- Part I A quick look at various zeta functions
- Part II Ihara zeta function and the graph theory prime number theorem
- Part III Edge and 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 Chebotarev density theorem
- 23 Siegel poles
- Part V Last look at the garden
- References
- Index
13 - Finite unramified coverings and Galois groups
from Part IV - Finite unramified Galois coverings of connected graphs
Published online by Cambridge University Press: 05 March 2013
- Frontmatter
- Contents
- List of illustrations
- Preface
- Part I A quick look at various zeta functions
- Part II Ihara zeta function and the graph theory prime number theorem
- Part III Edge and 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 Chebotarev density theorem
- 23 Siegel poles
- Part V Last look at the garden
- References
- Index
Summary
In this chapter we begin the study of Galois theory for finite unramified covering graphs. It leads to a generalization of Cayley and Schreier graphs and provides factorizations of zeta functions of normal coverings into products of Artin L-functions associated with representations of the Galois group of the covering. Coverings can also be used in constructions of Ramanujan graphs and in constructions of pairs of graphs that are isospectral but not isomorphic. Most of this chapter is taken from Stark and Terras [120]. Other references are Sunada [128] and Hashimoto [51]. Another theory of graph covering which is essentially equivalent can be found in Gross and Tucker [47]. Our coverings, however, differ in that we require all our graphs to be connected and in that our aim is to find analogs of the basic properties of finite-degree extensions of algebraic number fields and their zeta functions. It is also possible to consider infinite coverings such as the universal covering tree T of a finite graph X. We will not do so here except in passing. This is mostly a book about finite graphs, after all.
Definitions
If our graphs had no multiple edges or loops, our definition of covering would be Definition 2.9. If we want to prove the fundamental theorem of Galois theory for graphs with loops and multiple edges, Definition 2.9 will not be sufficient. We need to produce a more complicated definition of graph covering involving neighborhoods in directed graphs.
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- Chapter
- Information
- Zeta Functions of GraphsA Stroll through the Garden, pp. 105 - 116Publisher: Cambridge University PressPrint publication year: 2010