Characterizations of canonically compactifiable graphs via intrinsic metrics and algebraic properties

TL;DR

This paper characterizes canonically compactifiable graphs using intrinsic metrics and algebraic properties, establishing conditions for boundedness of finite energy functions and their algebraic structure.

Contribution

It provides a new characterization of canonically compactifiable graphs through intrinsic metrics and algebraic properties, answering recent open questions.

Findings

A graph is canonically compactifiable iff the underlying set is totally bounded under any finite measure intrinsic metric.

A graph is canonically compactifiable iff the space of finite energy functions forms an algebra.

The results connect energy form properties with metric and algebraic conditions on graphs.

Abstract

We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, Wojciechowski.