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Zeta functions of graphs. A stroll through the garden. (English) Zbl 1206.05003
Cambridge Studies in Advanced Mathematics 128. Cambridge: Cambridge University Press (ISBN 978-0-521-11367-0/hbk). xii, 239 p. £ 40.00; \$ 65.00 (2011).

The first part of the book is concerned with various types of zeta functions and their tentative connection with graph theory. The reader is reminded of Riemann’s, Dedekind’s and Epstein’s zeta functions as well as of Dirichlet’s and Artin’s $L$-functions. It continues with Ihara’s zeta function which is at the core of this book and the introduction of which deserves a little more attention.

Given a directed graph $G=\left(V,A\right)$, two cycles are called equivalent if they consist of the same arcs, in the same order but the start vertex may be different. The equivalence class of the cycle $C$ is denoted by $\left[C\right]$. The length of a cycle $C$ is denoted by $\nu \left(C\right)$ and is the number of its arcs. A backtrack of a cycle $C=\left({a}_{1},{a}_{2},\cdots ,{a}_{s}\right)$ is a subcycle of length 2 (i.e., ${a}_{j+1}={a}_{j}^{-1}$, for some $1\le j). A prime cycle is a backtrackless cycle which is not a power of another cycle. A prime of $G$ is an equivalence class $\left[P\right]$ of prime cycles.

Ihara’s zeta function for a finite, connected graph $G$ is defined for complex numbers $u$ of small modulus by:

${\zeta }_{G}\left(u\right)=\zeta \left(u,G\right)=\prod _{\left[P\right]}{\left(1-{u}^{\nu \left(P\right)}\right)}^{-1}$

where the product is extended over all primes $\left[P\right]$ of $G$.

Ihara’s main theorem, generalized by Bass, Hashimoto and others, provides a determinant formula for the computation of Ihara’s zeta function of a graph determinant which involves the adjacency matrix of the graph. The graph theory prime number theorem, stated here but proved in Part III of the book, evaluates the number $\pi \left(m\right)$ of primes $\left[P\right]$ whose length $\nu \left(P\right)$ equals $m$.

While Selberg’s zeta function which is described in chapter 3 looks less likely to have a natural correspondent in graph theory, Ruelle’s zeta function, which is covered next, is directly related to Ihara’s zeta function. Chapter 5 is a foray into chaos theory and its connections with the distribution of the zeros of the Ihara zeta function for regular graphs.

Part II starts with an extension of the Ihara zeta function to weighted graphs and continues with an adapted Riemann’s hypothesis for Ihara’s zeta function and with several functional equations for regular graphs. Regular Ramanujan graphs are covered as well. A proof of the graph theory prime number theorem concludes this part of the book.

Part III of the book contains Bass’s proof of Ihara’s determinant formula. It starts with the introduction of the edge zeta function, which is a multivariable complex function. The path zeta function, as introduced by Stark, is discussed next. The path zeta function exhibits, as well, a determinant formula.

It is worth mentioning that Ihara’s zeta function is a specialized version of the edge zeta function and the latter is a specialized form of the path zeta function.

Part IV of the book deals with the Galois theory for finite unramified coverings of connected graphs.

A covering of a finite graph is similar to an extension of an algebraic field or a function field. In this context, we are lead to factorizations of the zeta function of normal coverings as products of Artin $L$-functions associated with representations of the Galois group of the covering.

The concept of covering (adapted to some degree to make it workable) of an undirected finite graph is introduced in Chapter 13. A $d$-sheeted covering $Y/X$ is called normal or Galois when the order of the automorphism group of $Y/X$ is $d$.

The fundamental theorem of the Galois theory for unramified normal coverings of graphs is stated and proved in Chapter 14.

The book continues with the study of the behavior of primes in coverings, the Artin $L$-functions, the Frobenius automorphism and the construction of intermediate coverings using the Frobenius automorphism.

Artin $L$-functions come in several flavors and are covered extensively over several chapters. Besides the Ihara-Artin $L$-functions, we have the edge and path Artin $L$-functions.

Chapter 21 provides several examples of non-isomorphic, simple, regular graphs having the same Ihara zeta functions, as well.

Part IV concludes with the Chebotarev density theorem and with an overview of Siegel pole results.

Part V (and final) of the book reviews an application to error correcting codes, some chaos theory examples and ends with an open list of research problems.

The book is very appealing through its informal style and the variety of topics covered and may be considered the standard reference book in this field.

##### MSC:
 05-02 Research monographs (combinatorics) 05C25 Graphs and abstract algebra 11M26 Nonreal zeros of $\zeta \left(s\right)$ and $L\left(s,\chi \right)$; Riemann and other hypotheses 05C50 Graphs and linear algebra 11M06 $\zeta \left(s\right)$ and $L\left(s,\chi \right)$ 11M41 Other Dirichlet series and zeta functions 14G10 Zeta-functions and related questions
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