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In [[mathematics]], the '''Ihara zeta-function''' is a [[zeta function]] associated with a finite [[Graph (mathematics)|graph]]. It closely resembles the [[Selberg zeta function|Selberg zeta-function]], and is used to relate closed paths to the 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 (1985). 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.678</ref> | |||
==Definition== | |||
The Ihara zeta-function can be defined by a formula analogous to the Euler product for the [[Riemann zeta function]]: | |||
:<math>\frac{1}{\zeta_G(u)} = \prod_{p} ({1 - u^{L(p)}})</math> | |||
This product is taken over all prime walks ''p'' of the graph <math> G = (V, E) </math> - that is, closed cycles <math>p = (u_0, \cdots, u_{L(p)-1}, u_0)</math> such that | |||
:<math> (u_i, u_{(i+1)\bmod L(p)}) \in E~; \quad u_i \neq u_{(i+2) \bmod L(p)~}, </math> | |||
and <math> L(p) </math> is the length of cycle ''p'', as used in the formulae above.<ref>Terras (2010) p.12</ref> | |||
==Ihara's formula== | |||
Ihara showed that for regular graphs the zeta function is a rational function. If ''G'' is ''k''-regular with [[adjacency matrix]] ''A'' then<ref>Terras (1999) p.677</ref> | |||
:<math>\zeta_G(u) = \frac{(1-u^2)^{\chi(G)-1}}{\det(I - Au + (k-1)u^2I)} \ </math> | |||
where χ is the [[circuit rank]]. | |||
The Ihara zeta-function is in fact always the reciprocal of a polynomial: | |||
:<math>\zeta_G(u) = \frac{1}{\det (I-Tu)}~,</math> | |||
where ''T'' is 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.29</ref> | |||
== References == | |||
{{reflist}} | |||
* {{cite journal | first=Yasutaka | last=Ihara | title=On discrete subgroups of the two by two projective linear group over <math>{\mathfrak p}</math>-adic fields | journal=J. Math. Soc. Japan | volume=18 | year=1966 | pages=219–235 | zbl=0158.27702 }} | |||
* {{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=H. | last=Bass | authorlink=Hyman Bass | title=The Ihara-Selberg zeta function of a tree lattice | journal=International. J. Math.| volume=3 | year=1992 | pages=717–797 | doi=10.1142/S0129167X92000357 | issue=6 | zbl=0767.11025 }} | |||
* {{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. | editor4-last=Odlyzko| editor4-link=Andrew Odlyzko | publisher=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. | 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=0-521-11367-9 | zbl=1206.05003 }} | |||
[[Category:Zeta and L-functions]] | |||
[[Category:Algebraic graph theory]] | |||
{{algebra-stub}} | |||
{{numtheory-stub}} | |||
Revision as of 23:41, 9 July 2013
In mathematics, the Ihara zeta-function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta-function, and is used to relate closed paths to the spectrum of the adjacency matrix. The Ihara zeta-function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two 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 (1985). A regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis.[1]
Definition
The Ihara zeta-function can be defined by a formula analogous to the Euler product for the Riemann zeta function:
This product is taken over all prime walks p of the graph - that is, closed cycles such that
and is the length of cycle p, as used in the formulae above.[2]
Ihara's formula
Ihara showed that for regular graphs the zeta function is a rational function. If G is k-regular with adjacency matrix A then[3]
where χ is the circuit rank.
The Ihara zeta-function is in fact always the reciprocal of a polynomial:
where T is 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 groups, spectral graph theory, and dynamical systems, especially symbolic dynamics, where the Ihara zeta function is an example of a Ruelle zeta function.[4]
References
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
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