Editing Ihara zeta function

In [[mathematics]], the '''Ihara zeta function''' is a [[zeta function]] associated with a finite [[Graph (discrete mathematics)|graph]]. It closely resembles the [[Selberg zeta function]], and is used to relate closed walks to the [[Spectrum of a matrix|spectrum]] of the [[adjacency matrix]]. The Ihara zeta function was first defined by [[Yasutaka Ihara]] in the 1960s in the context of [[discrete group|discrete subgroups]] of the two-by-two [[p-adic number|p-adic]] [[special linear group]]. [[Jean-Pierre Serre]] suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was [[Toshikazu Sunada]] who put this suggestion into practice in 1985. As observed by Sunada, a [[regular graph]] is a [[Ramanujan graph]] if and only if its Ihara zeta function satisfies an analogue of the [[Riemann hypothesis]].<ref>Terras (1999) p.&nbsp;678</ref>

==Definition==
The Ihara zeta function is defined as the analytic continuation of the infinite product

:<math>\zeta_{G}(u)=\prod_{p}\frac{1}{1-u^{{L}(p)}},</math>

where ''L''(''p'') is the ''length'' <math>L(p)</math> of <math>p</math>. 
The product in the definition is taken over all prime [[closed geodesic]]s <math>p</math> of the graph <math>G = (V, E)</math>, where geodesics which differ by a [[circular shift|cyclic rotation]] are considered equal.  A ''closed geodesic'' <math>p</math> on <math>G</math> (known in graph theory as a "[[Cycle (graph theory)|reduced closed walk]]"; it is not a graph geodesic) is a finite sequence of vertices <math>p = (v_0, \ldots, v_{k-1})</math>  such that
:<math> (v_i, v_{(i+1)\bmod k}) \in E, </math>
:<math> v_i \neq v_{(i+2) \bmod k}. </math>
The integer <math>k</math> is the length <math>L(p)</math>. The closed geodesic <math>p</math> is ''prime'' if it cannot be obtained by repeating a closed geodesic <math>m</math> times, for an integer <math>m > 1</math>.

This graph-theoretic formulation is due to Sunada.

==Ihara's formula==
Ihara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function.
If <math>G</math> is a <math>q+1</math>-regular graph with [[adjacency matrix]] <math>A</math> then<ref>Terras (1999) p.&nbsp;677</ref>

:<math>\zeta_G(u) = \frac{1}{(1-u^2)^{r(G)-1}\det(I - Au + qu^2I)}, </math>

where <math>r(G)</math> is the [[circuit rank]] of <math>G</math>. If <math>G</math> is connected and has <math>n</math> vertices, <math>r(G)-1=(q-1)n/2</math>.

The Ihara zeta-function is in fact always the reciprocal of a [[graph polynomial]]:

:<math>\zeta_G(u) = \frac{1}{\det (I-Tu)}~,</math>

where <math>T</math> is Ki-ichiro Hashimoto's edge adjacency operator. [[Hyman Bass]] gave a determinant formula involving the adjacency operator.

==Applications==
The Ihara zeta function plays an important role in the study of [[free group]]s, [[spectral graph theory]], and [[dynamical systems]], especially [[symbolic dynamics]], where the Ihara zeta function is an example of a [[Ruelle zeta function]].<ref name=T29>Terras (2010) p.&nbsp;29</ref>

== References ==
{{reflist}}
* {{cite journal | first=Yasutaka | last=Ihara |authorlink=Yasutaka Ihara| title=On discrete subgroups of the two by two projective linear group over <math>{\mathfrak p}</math>-adic fields | journal=
Journal of the Mathematical Society of Japan | volume=18 | year=1966 | pages=219–235 | zbl=0158.27702|mr=0223463|doi=10.2969/jmsj/01830219 | doi-access=free }}
* {{cite book | first=Toshikazu | last=Sunada | authorlink=Toshikazu Sunada |title=Curvature and Topology of Riemannian Manifolds | volume=1201 | year=1986 | pages=266–284 | doi=10.1007/BFb0075662 | chapter=L-functions in geometry and some applications | series=[[Lecture Notes in Mathematics]] | isbn=978-3-540-16770-9 | zbl=0605.58046 }}
* {{cite journal | first=Hyman | last=Bass | authorlink=Hyman Bass | title=The Ihara-Selberg zeta function of a tree lattice | journal=
[[International Journal of Mathematics]]| volume=3 | year=1992 | pages=717–797 | doi=10.1142/S0129167X92000357 | issue=6 | zbl=0767.11025|mr=1194071 }}
* {{cite book | first=Harold M. | last=Stark | authorlink=Harold Stark | chapter=Multipath zeta functions of graphs | pages=601–615 | title=Emerging Applications of Number Theory | editor1-first=Dennis A. | editor1-last=Hejhal | editor1-link=Dennis Hejhal | editor2-first=Joel | editor2-last=Friedman | editor3-first=Martin C. | editor3-last=Gutzwiller | editor3-link=Martin Gutzwiller | editor4-first=Andrew M. |display-editors = 3 | editor4-last=Odlyzko| editor4-link=Andrew Odlyzko | publisher=[[Springer Science+Business Media|Springer]] | year=1999 | isbn=0-387-98824-6 | zbl=0988.11040 | series=IMA Vol. Math. Appl. | volume=109 }}
* {{cite book | first=Audrey | last=Terras | authorlink=Audrey Terras | chapter=A survey of discrete trace formulas | pages=643–681 | title=Emerging Applications of Number Theory | editor1-first=Dennis A. | editor1-last=Hejhal | editor1-link=Dennis Hejhal | editor2-first=Joel | editor2-last=Friedman | editor3-first=Martin C. | editor3-last=Gutzwiller | editor3-link=Martin Gutzwiller | editor4-first=Andrew M. |display-editors = 3 | editor4-last=Odlyzko| editor4-link=Andrew Odlyzko | publisher=Springer | year=1999 | isbn=0-387-98824-6 | zbl=0982.11031| series=IMA Vol. Math. Appl. | volume=109 }}
* {{cite book | title=Zeta Functions of Graphs: A Stroll through the Garden | volume=128 | series=Cambridge Studies in Advanced Mathematics | first=Audrey | last=Terras | authorlink=Audrey Terras | publisher=[[Cambridge University Press]] | year=2010 | isbn=978-0-521-11367-0 | zbl=1206.05003 }}

[[Category:Zeta and L-functions]]
[[Category:Algebraic graph theory]]