Difference between revisions of "Ihara zeta function"
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[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] | ||
| + | [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] | ||
Revision as of 00:49, 3 August 2014
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
