- Ihara zeta function
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 itsEuler characteristic . The Ihara zeta-function was first defined byYasutaka Ihara in the 1960s.Definition
The Ihara zeta-function can be defined by a formula analogous to the Euler product for the
Riemann zeta function ::frac{1}{zeta_G(u)} = prod_{p}(1-u^{L(p)})
This product is taken over all prime walks "p" of the graph G ,- that is, closed cycles p = (u_0, cdots, u_{L(p)-1}, u_0) such that
:u_i, u_{(i+1)mod L(p)}) in E~; quad u_i eq u_{(i+2) mod L(p)~},
and L(p) is the length of cycle "p".clarifyme
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.
Applications
The Ihara zeta-function plays an important role in the study of
free group s,spectral graph theory , anddynamical systems , especiallysymbolic dynamics .References
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