## Zeta Functions of Finite Graphs

J. Math. Sci. Univ. Tokyo
Vol. 7 (2000), No. 1, Page 7--25.

Kotani, Motoko ; Sunada, Toshikazu
Zeta Functions of Finite Graphs
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Abstract:
Poles of the {\it Ihara zeta function} associated with a finite graph are described by graph-theoretic quantities. Elementary proofs based on the notions of {\it oriented line graphs}, {\it Perron-Frobenius operators}, and {\it discrete Laplacians} are provided for Bass's theorem on the determinant expression of the zeta function and Hashimoto's theorems on the pole at $u=1$.

Keywords: zeta functions, graphs, oriented line graph, adjacency operators

Mathematics Subject Classification (2000): 68R10, 05C50, 14G10
Mathematical Reviews Number: MR1749978