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  "body": "\n## Statement\n\nIhara (and Sunada in the graph-theoretic setting) showed that for regular graphs the zeta function is a rational function. If $G$ is a $q+1$-regular graph with [[adjacency matrix|08B9CX109G]] $A$ then\n\n$$\\zeta_G(u) = \\frac{1}{(1-u^2)^{r(G)-1}\\det(I - Au + qu^2I)},$$\n\nwhere $r(G)$ is the circuit rank of $G$. If $G$ is connected and has $n$ vertices, $r(G)-1=(q-1)n/2$.\n\nThe Ihara zeta-function is in fact always the reciprocal of a graph polynomial:\n\n$$\\zeta_G(u) = \\frac{1}{\\det (I-Tu)}~,$$\n\nwhere $T$ is Ki-ichiro Hashimoto's edge adjacency operator. Hyman Bass gave a determinant formula involving the adjacency operator.\n",
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