Difference between revisions of "Ihara zeta function"
From specialfunctionswiki
(Created page with "Let $X$ be a finite graph. The Ihara zeta function is given by the formula $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ where $[C]$ is the set of equiva...") |
|||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
Let $X$ be a finite [[graph]]. The Ihara zeta function is given by the formula | Let $X$ be a finite [[graph]]. The Ihara zeta function is given by the formula | ||
$$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ | $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ | ||
| − | where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the Euler product representation of the [[Riemann zeta function]]. | + | where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the [[Euler product]] representation of the [[Riemann zeta function]]. |
=References= | =References= | ||
| − | [http://math.tamu.edu/~grigorch/publications/zukiharanewbisk2.ps The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps - Rostislav I. Grigorchuk, Andrzej Zuk] | + | [http://math.tamu.edu/~grigorch/publications/zukiharanewbisk2.ps The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps - Rostislav I. Grigorchuk, Andrzej Zuk]<br /> |
| + | [http://math.ucsd.edu/~aterras/snowbird.pdf What are zeta functions of graphs and what are they good for? - Matthew D. Horton, H.M. Stark, Audrey A. Terras] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 18:52, 24 May 2016
Let $X$ be a finite graph. The Ihara zeta function is given by the formula $$\zeta_X(t) = \displaystyle\prod_{[C]} \dfrac{1}{1-t^{|C|}},$$ where $[C]$ is the set of equivalence classes of primitive closed paths $C$ in $X$ and $|C|$ denotes the length of $C$. This formula is an analogue of the Euler product representation of the Riemann zeta function.
References
The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps - Rostislav I. Grigorchuk, Andrzej Zuk
What are zeta functions of graphs and what are they good for? - Matthew D. Horton, H.M. Stark, Audrey A. Terras
