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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.
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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