A New Proof of Bartholdi's Theorem
Hirobumi Mizuno
and Iwao Sato
dagger
DOI: 10.1007/s10801-005-4526-3
Abstract
Pages: 259–271
Keywords: keywords zeta function; graph; cycle; bump
Full Text: PDF
References
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2. L. Bartholdi, “Counting paths in graphs,” Enseign. Math. 45 (1999), 83-131.
3. H. Bass, “The Ihara-Selberg zeta function of a tree lattice,” Internat. J. Math. 3 (1992), 717-797.
4. R. Bowen and O. Lanford, “Zeta functions of restrictions of the shift transformation,” Proc. Symp. Pure Math. 14 (1970), 43-50.
5. H. Davenport, Multiplicative Number Theory, Springer-Verlag, New York, 1981.
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8. K. Hashimoto, “Zeta Functions of Finite Graphs and Representations of p-Adic Groups,” Adv. Stud. Pure Math. Vol. 15, pp. 211-280, Academic Press, New York, 1989.
9. K. Hashimoto, “On the zetaand L-functions of finite graphs,” Internat. J. Math. 1 (1990), 381-396.
10. K. Hashimoto, “Artin-type L-functions and the density theorem for prime cycles on finite graphs,” Internat. J. Math. 3 (1992), 809-826.
11. Y. Ihara, “On discrete subgroups of the two by two projective linear group over p-adic fields,” J. Math. Soc. Japan 18 (1966), 219-235.
12. M. Kotani and T. Sunada, “Zeta functions of finite graphs,” J. Math. Sci. U. Tokyo 7 (2000), 7-25.
13. S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, MA, 1970.
14. A. Lubotzky, R. Phillips, and P. Sarnak, “Ramanujan graphs,” Combinatorica 8 (3) (1988), 261-277.
15. S. Northshield, “A note on the zeta function of a graph,” J. Combin. Theory Ser. B 74 (1998), 408-410.
16. D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval, CRM Monograph Series Vol. 4, Amer. Math. Soc., Providence, RI, 1994.
17. A. Selberg,“Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,” J. Indian Math. 20 (1956), 47-87.
18. J.-P. Serre, Trees, Springer-Verlag, New York, 1980.
19. H.M. Stark and A.A. Terras, “Zeta functions of finite graphs and coverings,” Adv. Math. 121 (1996), 124-165.
20. T. Sunada, “L-Functions in Geometry and Some Applications,” in “Lecture Notes in Math.” Vol. 1201, pp. 266-284, Springer-Verlag, New York, 1986.
21. T. Sunada, “Fundamental Groups and Laplacians” (in Japanese), Kinokuniya, Tokyo, 1988.
22. A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge Univ. Press, Cambridge (1999).
23. A.B. Venkov and A.M. Nikitin, “The Selberg trace formula, Ramanujan graphs and some problems of mathematical physics,” St. Petersburg Math. J. 5(3) (1994), 419-484.
2. L. Bartholdi, “Counting paths in graphs,” Enseign. Math. 45 (1999), 83-131.
3. H. Bass, “The Ihara-Selberg zeta function of a tree lattice,” Internat. J. Math. 3 (1992), 717-797.
4. R. Bowen and O. Lanford, “Zeta functions of restrictions of the shift transformation,” Proc. Symp. Pure Math. 14 (1970), 43-50.
5. H. Davenport, Multiplicative Number Theory, Springer-Verlag, New York, 1981.
6. D. Foata and D. Zeilberger, “A combinatorial proof of Bass's evaluations of the Ihara-Selberg zeta function for graphs,” Trans. Amer. Math. Soc. 351 (1999), 2257-2274.
7. R.I. Grigorchuk, “Symmetrical random walks on discrete groups,” in “Adv. Probab. Rel. Top.” D. Griffeath, (Ed.) Vol. 6, M. Dekker 1980, 285-325, pp. 132-152.
8. K. Hashimoto, “Zeta Functions of Finite Graphs and Representations of p-Adic Groups,” Adv. Stud. Pure Math. Vol. 15, pp. 211-280, Academic Press, New York, 1989.
9. K. Hashimoto, “On the zetaand L-functions of finite graphs,” Internat. J. Math. 1 (1990), 381-396.
10. K. Hashimoto, “Artin-type L-functions and the density theorem for prime cycles on finite graphs,” Internat. J. Math. 3 (1992), 809-826.
11. Y. Ihara, “On discrete subgroups of the two by two projective linear group over p-adic fields,” J. Math. Soc. Japan 18 (1966), 219-235.
12. M. Kotani and T. Sunada, “Zeta functions of finite graphs,” J. Math. Sci. U. Tokyo 7 (2000), 7-25.
13. S. Lang, Algebraic Number Theory, Addison-Wesley, Reading, MA, 1970.
14. A. Lubotzky, R. Phillips, and P. Sarnak, “Ramanujan graphs,” Combinatorica 8 (3) (1988), 261-277.
15. S. Northshield, “A note on the zeta function of a graph,” J. Combin. Theory Ser. B 74 (1998), 408-410.
16. D. Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval, CRM Monograph Series Vol. 4, Amer. Math. Soc., Providence, RI, 1994.
17. A. Selberg,“Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series,” J. Indian Math. 20 (1956), 47-87.
18. J.-P. Serre, Trees, Springer-Verlag, New York, 1980.
19. H.M. Stark and A.A. Terras, “Zeta functions of finite graphs and coverings,” Adv. Math. 121 (1996), 124-165.
20. T. Sunada, “L-Functions in Geometry and Some Applications,” in “Lecture Notes in Math.” Vol. 1201, pp. 266-284, Springer-Verlag, New York, 1986.
21. T. Sunada, “Fundamental Groups and Laplacians” (in Japanese), Kinokuniya, Tokyo, 1988.
22. A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge Univ. Press, Cambridge (1999).
23. A.B. Venkov and A.M. Nikitin, “The Selberg trace formula, Ramanujan graphs and some problems of mathematical physics,” St. Petersburg Math. J. 5(3) (1994), 419-484.