Ihara's Formula
Ihara showed that for regular graphs the zeta function is a rational function. If G is k-regular with adjacency matrix A then
where χ is the circuit rank.
The Ihara zeta-function is in fact always the reciprocal of a polynomial:
where T is Hashimoto's edge adjacency operator. Hyman Bass gave a determinant formula involving the adjacency operator.
Read more about this topic: Ihara Zeta Function
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