Note on intrinsic metrics on graphs

TL;DR

This paper investigates the set of intrinsic metrics on graphs, showing that only finite star graphs have a largest intrinsic metric and providing characterizations for certain symmetric infinite graphs.

Contribution

It characterizes when graphs admit a largest intrinsic metric and describes properties of intrinsic metrics on various classes of graphs.

Findings

Only finite star graphs admit a largest intrinsic metric.

Infinite locally finite graphs do not admit a largest intrinsic metric.

Characterization of intrinsic metrics with finite balls on weakly spherically symmetric graphs.

Abstract

We study the set of intrinsic metrics on a given graph. This is a convex compact set and it carries a natural order. We investigate existence of largest elements with respect to this order. We show that the only locally finite graphs which admit a largest intrinsic metric are certain finite star graphs. In particular all infinite locally finite graphs do not admit a largest intrinsic metric. Moreover, we give a characterization for the existence of intrinsic metrics with finite balls for weakly spherically symmetric graphs.