On the Ihara expression for the generalized weighted zeta function

TL;DR

This paper extends the Ihara expression to the generalized weighted zeta function of finite digraphs, including those with multi-arcs and loops, providing a determinant formula for these complex structures.

Contribution

It introduces a new Ihara expression for the generalized weighted zeta function applicable to digraphs with multi-arcs and loops, expanding the scope of existing formulas.

Findings

Derived a determinant expression for the generalized weighted zeta function

Extended Ihara's formula to digraphs with multi-arcs and loops

Confirmed the applicability of the formula to complex digraph structures

Abstract

We consider the generalized weighted zeta function for a finite digraph, and show that it has the Ihara expression, a determinant expression of graph zeta functions, with a certain specified definition for inverse arcs. A finite digraph in this paper allows multi-arcs or multi-loops.