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:

\zeta_{G}(u)^{-1} = \prod_{p}(1-u^{d_{p}})

This product is taken over all prime walks p of the graph G ,- that is, closed paths p = (u_1, \cdots, u_k, u_1) such that

(u_i, u_{(i+1) \mod k}) \in E~; \quad u_i \neq u_{(i+2) \mod k~},-

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:

\zeta_G(u) = \frac{1}{\det (I-Tu)}~,

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 {\mathfrak p}-adic fields, J. Math. Soc. Japan 18 1966 219--235
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