Ihara zeta function
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In mathematics, the Ihara zeta-function closely resembles the Selberg zeta-function, and is used to relate the spectrum of the adjacency matrix of a graph G = (V,E) to its Euler characteristic. The Ihara zeta-function was first defined by Yasutaka Ihara in the 1960-s.
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[edit] 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 G ,- that is, closed paths 
such that
and dp is the length of p (k in the notation of the previous remark).
[edit] Ihara's formula
The Ihara zeta-function is in fact always the reciprocal of a polynomial:
where T is Hashimoto's edge adjacency operator.
[edit] Applications
The Ihara zeta-function plays an important role in the study of free groups, spectral graph theory, and dynamical systems, especially symbolic dynamics.
[edit] References
- Y. Ihara, On discrete subgroups of the two by two projective linear group over
-adic fields, J. Math. Soc. Japan 18 1966 219--235
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