The scattering matrix with respect to an Hermitian matrix of a graph

TL;DR

This paper provides a new proof and generalization of Gnutzmann and Smilansky's formula for the scattering matrix of a graph relative to a Hermitian matrix, introduces an $L$-function of a graph, and relates it to graph coverings.

Contribution

It offers a novel proof technique, extends the formula to regular graph coverings, and defines an $L$-function with a determinant expression, linking it to the scattering matrix.

Findings

New proof of Gnutzmann and Smilansky's formula using zeta function techniques

Generalization of the formula to regular coverings of graphs

Definition of an $L$-function for graphs with a determinant expression

Abstract

Recently, Gnutzmann and Smilansky presented a formula for the bond scattering matrix of a graph with respect to a Hermitian matrix. We present another proof for this Gnutzmann and Smilansky's formula by a technique used in the zeta function of a graph. Furthermore, we generalize Gnutzmann and Smilansky's formula to a regular covering of a graph. Finally, we define an $L$-fuction of a graph, and present a determinant expression. As a corollary, we express the generalization of Gnutzmann and Smilansky's formula to a regular covering of a graph by using its $L$-functions.