The $p$-harmonic boundary and $D_p$-massive subsets of a graph of bounded degree

TL;DR

This paper explores the relationship between the $p$-harmonic boundary and $D_p$-massive subsets of bounded degree graphs, establishing a correspondence between the number of disjoint $D_p$-massive subsets and the size of the $p$-harmonic boundary.

Contribution

It proves a bidirectional link between the number of disjoint $D_p$-massive subsets and the cardinality of the $p$-harmonic boundary in graphs of bounded degree.

Findings

Number of disjoint $D_p$-massive subsets bounds the size of the $p$-harmonic boundary.

The converse statement also holds, linking boundary size to $D_p$-massive subsets.

Establishes a fundamental connection in potential theory on graphs.

Abstract

Let $p$ be a real number greater than one and let $\Gamma$ be a graph of bounded degree. We investigate links between the $p$-harmonic boundary of $\Gamma$ and the $D_p$-massive subsets of $\Gamma$. In particular, if there are $n$ pairwise disjoint $D_p$-massive subsets of $\Gamma$, then the $p$-harmonic boundary of $\Gamma$ consists of at least $n$ elements. We also show that the converse of this statement is also true.