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 its Euler characteristic. The Ihara zeta-function was first defined by Yasutaka 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 groups, spectral graph theory, and dynamical systems, especially symbolic dynamics.

References

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