The quaternionic weighted zeta function of a graph

TL;DR

This paper introduces a quaternionic weighted zeta function for graphs, extending the Ihara zeta function, and provides determinant expressions crucial for graph zeta function theory.

Contribution

It defines a new quaternionic zeta function for graphs and derives its determinant expressions, extending existing graph zeta function frameworks.

Findings

Defined a quaternionic weighted zeta function for graphs

Derived two Study determinant expressions for the zeta function

Extended the Ihara zeta function to quaternionic weights

Abstract

We establish the quaternionic weighted zeta function of a graph and its Study determinant expressions. For a graph with quaternionic weights on arcs, we define a zeta function by using an infinite product which is regarded as the Euler product. This is a quaternionic extension of the square of the Ihara zeta function. We show that the new zeta function can be expressed as the exponential of a generating function and that it has two Study determinant expressions, which are crucial for the theory of zeta functions of graphs.