Some results concerning the $p$-Royden and $p$-harmonic boundaries of a graph of bounded degree

TL;DR

This paper investigates the structure of the $p$-Royden and $p$-harmonic boundaries of bounded degree graphs, revealing their topological properties and characterizing the $p$-harmonic boundary through infinite paths.

Contribution

It provides a topological characterization of the $p$-Royden boundary and describes the $p$-harmonic boundary in terms of extreme points of path sets, advancing boundary theory in graph analysis.

Findings

The $p$-Royden boundary minus the $p$-harmonic boundary is an $F_{\sigma}$-set.

The $p$-harmonic boundary is characterized by extreme points of certain path subsets.

The results apply to connected graphs with bounded degree.

Abstract

Let $p$ be a real number greater than one and let $\Gamma$ be a connected graph of bounded degree. We show that the $p$-Royden boundary of $\Gamma$ with the $p$-harmonic boundary removed is a $F_{\sigma}$-set. We also characterize the $p$-harmonic boundary of $\Gamma$ in terms of the intersection of the extreme points of a certain subset of one-sided infinite paths in $\Gamma$.