Editing Ihara zeta function (section)

==Ihara's formula==
Ihara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function.
If <math>G</math> is a <math>q+1</math>-regular graph with [[adjacency matrix]] <math>A</math> then<ref>Terras (1999) p.&nbsp;677</ref>

:<math>\zeta_G(u) = \frac{1}{(1-u^2)^{r(G)-1}\det(I - Au + qu^2I)}, </math>

where <math>r(G)</math> is the [[circuit rank]] of <math>G</math>. If <math>G</math> is connected and has <math>n</math> vertices, <math>r(G)-1=(q-1)n/2</math>.

The Ihara zeta-function is in fact always the reciprocal of a [[graph polynomial]]:

:<math>\zeta_G(u) = \frac{1}{\det (I-Tu)}~,</math>

where <math>T</math> is Ki-ichiro Hashimoto's edge adjacency operator. [[Hyman Bass]] gave a determinant formula involving the adjacency operator.