The quaternionic second weighted zeta function of a graph and the Study determinant

TL;DR

This paper introduces a quaternionic generalization of the second weighted zeta function of a graph, utilizing the Study determinant to extend classical determinant expressions to quaternionic matrices.

Contribution

It defines the quaternionic second weighted zeta function using the Study determinant and derives its Bass type determinant expression and Euler product.

Findings

Established a quaternionic analogue of the Ihara zeta function

Derived the Study determinant expression of Bass type

Presented the Euler product for the quaternionic zeta function

Abstract

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 Study. 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.