Editing Ihara zeta function (section)

==Definition==
The Ihara zeta function is defined as the analytic continuation of the infinite product

:<math>\zeta_{G}(u)=\prod_{p}\frac{1}{1-u^{{L}(p)}},</math>

where ''L''(''p'') is the ''length'' <math>L(p)</math> of <math>p</math>. 
The product in the definition is taken over all prime [[closed geodesic]]s <math>p</math> of the graph <math>G = (V, E)</math>, where geodesics which differ by a [[circular shift|cyclic rotation]] are considered equal.  A ''closed geodesic'' <math>p</math> on <math>G</math> (known in graph theory as a "[[Cycle (graph theory)|reduced closed walk]]"; it is not a graph geodesic) is a finite sequence of vertices <math>p = (v_0, \ldots, v_{k-1})</math>  such that
:<math> (v_i, v_{(i+1)\bmod k}) \in E, </math>
:<math> v_i \neq v_{(i+2) \bmod k}. </math>
The integer <math>k</math> is the length <math>L(p)</math>. The closed geodesic <math>p</math> is ''prime'' if it cannot be obtained by repeating a closed geodesic <math>m</math> times, for an integer <math>m > 1</math>.

This graph-theoretic formulation is due to Sunada.