Cauchy Transforms of Colored Graphs in Two Variables

TL;DR

The paper develops a two-variable Nevanlinna framework for rational inner Pick functions arising from colored graphs, linking analytic representations to graph structure via $f_G(z,w) = \left\langle \mathcal{A}_G^{-1} e_k, e_k\right\rangle$. It shows that graph operations have crisp algebraic effects on the representing functions: star products yield additive laws on reciprocals $g_G = 1/f_G$ (via Schur complements), while comb products induce composition-like relations (reductions to loops). A central result connects boundary regularity to graph distance: for a single $w$-colored vertex, the order of contact at $(\infty,0)$ equals twice the shortest path length to that vertex, enabling design of functions with prescribed boundary behavior. The work also includes a combinatorial aside on stick graphs and raises open questions about multi-$w$ configurations and broader colorings, suggesting rich interactions between graph structure, operator models, and boundary analytic phenomena.

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.

Figures (7)

• Figure 1: Notation for coloring vertices, describes adding self-loops with weights $-z$ or $-w$.
• Figure 2: Example of a star product of two graphs.
• Figure 3: Example of a colored graph reduction.
• Figure 4: Example of a comb product of two graphs.
• Figure 5: Example of a $z$-compatible comb product of two graphs.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2: pascoe1bps2bickelhong
• Lemma 2.1: aty12 Proposition 2.2
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof