Cauchy Transforms of Colored Graphs in Two Variables

TL;DR

This paper explores how the Cauchy transforms of colored graphs in two variables relate to their structure, providing new relations and insights into their boundary behavior and regularity.

Contribution

It introduces relations between representing functions of graph products via Schur complements and analyzes the impact of graph structure on boundary regularity.

Findings

Derived relations between representing functions of graph products.

Connected graph structure to boundary regularity of functions.

Provided insights into boundary singularities and their regularity.

Abstract

By designating vertices with variables, a simple undirected graph can be augmented to have an associated representing rational function in two variables taking the complex bi-upper halfplane to itself. We give relations between representing functions of certain products of such graphs by way of Schur complements. We also study the connection between the structure of the graph and the regularity of the representing function at a boundary singularity.