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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
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