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The quaternionic second weighted zeta function of a graph and the Study determinant. (English) Zbl 1352.05115

Summary: We establish a generalization of the second weighted zeta function of a graph to the case of quaternions. For an arc-weighted graph whose weights are quaternions, we define the second weighted zeta function by using the Study determinant that is a quaternionic determinant for quaternionic matrices defined by E. Study [Acta Math. 42, 1–61 (1918; JFM 46.0144.06)]. This definition is regarded as a quaternionic analogue of the determinant expression of Hashimoto type for the Ihara zeta function of a graph. We derive the Study determinant expression of Bass type and the Euler product for the quaternionic second weighted zeta function.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A15 Determinants, permanents, traces, other special matrix functions
11R52 Quaternion and other division algebras: arithmetic, zeta functions

Citations:

JFM 46.0144.06
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References:

[1] Aslaksen, H., Quaternionic determinants, Math. Intelligencer, 18, 3, 57-65 (1996) · Zbl 0881.15007
[2] Bass, H., The Ihara-Selberg zeta function of a tree lattice, Internat. J. Math., 3, 717-797 (1992) · Zbl 0767.11025
[3] Berstel, J.; Reutenauer, C., Noncommutative Rational Series with Applications (2011), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1250.68007
[4] Foata, D.; Zeilberger, D., A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs, Trans. Amer. Math. Soc., 351, 2257-2274 (1999) · Zbl 0921.05025
[5] Hashimoto, K., Zeta Functions of Finite Graphs and Representations of \(p\)-Adic Groups, Adv. Stud. Pure Math., vol. 15, 211-280 (1989), Academic Press: Academic Press New York
[6] Hashimoto, K., On zeta and L-functions of finite graphs, Internat. J. Math., 1, 381-396 (1990) · Zbl 0734.14008
[7] Higuchi, H.; Konno, N.; Sato, I.; Segawa, E., A note on the discrete-time evolutions of quantum walk on a graph, J. Math-for-Ind., 5, 2013B-3, 103-109 (2013) · Zbl 1301.81066
[8] Ihara, Y., On discrete subgroups of the two by two projective linear group over \(p\)-adic fields, J. Math. Soc. Japan, 18, 219-235 (1966) · Zbl 0158.27702
[9] Konno, N., Quaternionic quantum walks, Quantum Stud.: Math. Found., 2, 63-76 (2015) · Zbl 1317.81184
[10] Konno, N.; Mitsuhashi, H.; Sato, I., The discrete-time quaternionic quantum walk on a graph, Quantum Inf. Process., 15, 651-673 (2016) · Zbl 1333.81210
[11] Konno, N.; Sato, I., On the relation between quantum walks and zeta functions, Quantum Inf. Process., 11, 2, 341-349 (2012) · Zbl 1241.81045
[12] Kotani, M.; Sunada, T., Zeta functions of finite graphs, J. Math. Sci. Univ. Tokyo, 7, 7-25 (2000) · Zbl 0978.05051
[13] Lothaire, M., Combinatorics on Words (1997), Cambridge University Press · Zbl 0874.20040
[14] Mizuno, H.; Sato, I., Weighted zeta functions of graphs, J. Combin. Theory Ser. B, 91, 169-183 (2004) · Zbl 1048.05044
[15] Mizuno, H.; Sato, I., The scattering matrix of a graph, Electron. J. Combin., 15, Article R96 pp. (2008) · Zbl 1163.05326
[16] Reutenauer, C.; Schützenberger, M-P., A formula for the determinant of a sum of matrices, Lett. Math. Phys., 13, 299-302 (1987) · Zbl 0628.15005
[17] Sato, I., A new Bartholdi zeta function of a graph, Int. J. Algebra, 1, 269-281 (2007) · Zbl 1126.05070
[18] Serre, J.-P., Trees (1980), Springer-Verlag: Springer-Verlag New York
[19] Stark, H. M.; Terras, A. A., Zeta functions of finite graphs and coverings, Adv. Math., 121, 124-165 (1996) · Zbl 0874.11064
[20] Study, E., Zur Theorie der lineare Gleichungen, Acta Math., 42, 1-61 (1920) · JFM 46.0144.06
[21] Smilansky, U., Quantum chaos on discrete graphs, J. Phys. A: Math. Theor., 40, F621-F630 (2007) · Zbl 1124.81024
[22] Sunada, T., \(L\)-Functions in Geometry and Some Applications, Lecture Notes in Math., vol. 1201, 266-284 (1986), Springer-Verlag: Springer-Verlag New York
[23] Sunada, T., Fundamental Groups and Laplacians (1988), Kinokuniya: Kinokuniya Tokyo, (in Japanese) · Zbl 0646.58027
[24] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57 (1997) · Zbl 0873.15008
[25] Zhang, F., Matrix Theory (2011), Springer: Springer New York
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